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Parametric equation

A parametric equation is a mathematical expression that defines a curve, surface, or higher-dimensional object in terms of one or more independent variables known as parameters, typically allowing the coordinates to be written as explicit functions of these parameters rather than relating them directly to each other.^[1] For instance, in two dimensions, a parametric curve is commonly represented by the pair of equations x = f(t) and y = g(t), where t is the parameter varying over an interval, generating points (x, y) that trace the curve.^[2] This approach is particularly useful for describing paths that are not functions of a single variable, such as closed loops or self-intersecting shapes.^[3] Parametric equations offer flexibility in parameterization, as the same curve can be expressed using different parameters or even multiple parameters for surfaces, such as x = f(u, v), y = g(u, v), and z = h(u, v) in three dimensions.^[1] Classic examples include the circle, parameterized as x = r \cos t and y = r \sin t for t \in [0, 2\pi], which traces the full circumference as t varies.^[3] Another notable case is the cycloid, the path traced by a point on the rim of a rolling circle, given by x = a(t - \sin t) and y = a(1 - \cos t), where a is the circle's radius; this curve appears in applications like the motion of a wheel or pendulum arcs.^[2] In applications, parametric equations model dynamic phenomena, such as planetary orbits—Earth's elliptical path around the Sun can be parameterized by time t in days over approximately 365 units—or projectile motion under gravity.^[3] They also facilitate computations like arc length or tangent vectors by differentiating with respect to the parameter, and the parameter can often be eliminated to obtain a Cartesian equation, as in converting x = t^2 and y = t to y^2 = x, revealing a parabola.^[2] This versatility extends to fields like computer graphics, physics simulations, and engineering design, where parametric representations enable efficient plotting and analysis.^[1]

Fundamentals

Definition and General Form

A parametric equation expresses one or more dependent variables as functions of one or more independent parameters, allowing for the description of curves or surfaces in a flexible manner.^[4] In the basic case for plane curves, the coordinates are defined as x = f(t) and y = g(t), where t is the parameter and f and g are real-valued functions.^[5] The resulting parametric curve consists of all points (f(t), g(t)) as t varies over its domain, typically an interval or subset of the real numbers.^[5] For curves in three-dimensional space, the formulation extends to x = f(t), y = g(t), and z = h(t), where h is another function of t.^[6] In vector notation, this is compactly written as \vec{r}(t) = \langle x(t), y(t), z(t) \rangle, representing the position vector from the origin to a point on the curve.^[6] The domain of t specifies the portion of the curve traced, such as t \in [0, 2\pi] for a full period or t \in \mathbb{R} for an infinite extent.^[5] Parametric equations can be presented in scalar form, with separate equations for each coordinate, or in the unified vector form for brevity and clarity in higher dimensions.^[7] They may also be defined piecewise, using different functional expressions over disjoint subintervals of the parameter's domain to describe more complex or non-smooth paths.^[8] This approach assumes familiarity with functions and domains in calculus, where the parameter t maps from its domain to the coordinate values.^[5] A representative example is the straight line passing through the origin with slope 2, given by the scalar parametric equations x = t, y = 2t for t \in \mathbb{R}, or in vector form \vec{r}(t) = \langle t, 2t \rangle.^[5] As t increases, the point (x, y) moves linearly along the line y = 2x.^[5]

Historical Development

The origins of parametric equations can be traced to ancient Greek geometry, where mechanical constructions foreshadowed parametric representations. Around 430 BCE, the sophist and mathematician Hippias of Elis invented the quadratrix, a curve designed to trisect angles, generated by the intersection of a line rotating uniformly from a fixed point and another line descending parallel to the y-axis at constant speed; this construction is recognized as a precursor to parametric curves due to its time-dependent definition.^[9] The quadratrix marked one of the earliest attempts to describe non-circular curves through variable motion, bridging geometric construction and functional description. The 17th century brought foundational advancements with the emergence of analytic geometry. René Descartes, in his 1637 work La Géométrie, and Pierre de Fermat, through unpublished manuscripts from the 1630s, independently developed coordinate systems that allowed curves to be expressed algebraically, setting the stage for parametric formulations by relating points to parameters like time or angle. Isaac Newton further propelled this evolution in his 1671 manuscript De methodis serierum et fluxionum, introducing the "organic description" of curves—a parametric method using two rulers, one rotating about a fixed point and the other sliding along a fixed line, to generate algebraic curves of arbitrary degree through proportional motion. Newton's approach emphasized constructive generation, influencing subsequent mechanical and analytical treatments of curves. In the 19th century, parametric representations gained formal rigor in higher geometry and analysis. Julius Plücker, a German mathematician, introduced line coordinates in his 1833 book System der analytischen Geometrie, parametrizing lines in three-dimensional projective space with six homogeneous coordinates derived from two points on the line, which provided a dual to point coordinates and advanced the parametric study of linear elements. Karl Weierstrass, known as the father of modern analysis, emphasized parametric forms in his Berlin lectures from the 1860s onward, particularly through the Weierstrass-Enneper parameterization, which expressed minimal surfaces via holomorphic functions of a complex parameter, integrating parametric equations into differential geometry and function theory. The 20th century integrated parametric equations into algebraic geometry and computational fields. David Hilbert's 1893 Nullstellensatz established a correspondence between polynomial ideals and algebraic varieties over algebraically closed fields, with implications for parametric solutions by clarifying conditions under which systems of equations admit rational parametrizations, thus supporting the study of birational maps and curve parametrization in algebraic geometry. In computer-aided geometric design (CAGD), parametric equations became essential for modeling; Pierre Bézier developed parametric polynomial curves in the 1960s at Renault, using control points to define smooth paths for automotive surfaces, which popularized rational parametric representations in engineering and graphics.^[10] Prominent figures shaped this trajectory: Newton (1643–1727), whose organic methods unified kinematics and algebra in curve generation; Plücker (1801–1868), whose coordinates extended parametrization to projective lines, influencing Grassmann and Klein; and Bézier (1910–1999), whose CAGD innovations made parametric forms practical for computational design.^[10]

Advantages Over Implicit and Explicit Forms

Parametric equations offer significant flexibility in representing curves that cannot be easily expressed as explicit functions y = f(x) or implicit equations F(x, y) = 0, particularly for multi-valued relations. For instance, a circle cannot be fully captured by an explicit form like y = \pm \sqrt{r^2 - x^2} without branching, as it fails to describe the complete closed path in a single expression, whereas the parametric form x = r \cos t, y = r \sin t traces the entire curve smoothly as t varies from 0 to $2\pi.^[11] This approach avoids the limitations of explicit representations for non-functions, such as ellipses or more complex conics, enabling a unified description without algebraic manipulation.^[12] A key advantage lies in the natural incorporation of a parameter t that can represent time, arc length, or another progression metric, facilitating the study of dynamic properties like motion and velocity. In kinematics, for an object following \vec{r}(t) = (x(t), y(t)), the velocity vector is directly \vec{v}(t) = \frac{d\vec{r}}{dt} = \left( \frac{dx}{dt}, \frac{dy}{dt} \right), providing both magnitude and direction that are straightforward to compute and interpret, unlike deriving them from static implicit or explicit forms which often require additional steps such as implicit differentiation.^[5] This parameterization is especially useful for simulating trajectories, such as projectile motion parameterized by time t, where explicit forms might obscure the temporal evolution.^[11] Parametric representations excel in handling piecewise and non-algebraic curves, such as splines or those involving transcendental functions like sine and cosine in cycloids, which are cumbersome or impossible to express explicitly without piecewise definitions. For example, B-spline curves use parametric forms to ensure local control and smoothness, allowing modifications to one segment without affecting others, a feature not readily available in implicit equations that treat the entire curve holistically.^[13] Computationally, parametric equations avoid the need to solve for inverses or eliminate parameters iteratively, making them ideal for numerical simulations and graphics where t advances sequentially to generate points efficiently, independent of dimension or slope issues that plague explicit plotting.^[12] While powerful, parametric forms are not without drawbacks, such as non-uniqueness in parameterization (multiple parameter choices for the same curve) and the occasional need for rational functions to achieve exact algebraic representations, like in conic sections.^[14] Nonetheless, these advantages make parametric equations preferable in applications requiring adaptability, dynamics, and computational tractability over the rigidity of implicit or explicit alternatives.

Parametric Curves in Euclidean Space

Plane Curves (2D)

Parametric equations provide a method to describe curves in the plane by expressing the coordinates x and y as functions of a parameter t, typically over an interval [a, b], in the form x = x(t) and y = y(t).^[5] The curve consists of the set of points (x(t), y(t)) traced as t varies within the interval, allowing representation of curves that may not be easily expressed as a single equation relating x and y directly.^[15] If the functions x(t) and y(t) are continuous and differentiable, the parametric form captures the path and its geometric properties without restricting the curve to a function of one variable.^[5] To obtain an implicit relation between x and y, the parameter t can be eliminated, yielding an equation F(x, y) = 0 that defines the curve algebraically, provided such an elimination is possible.^[16] This process is not always straightforward and may involve solving for t from one equation and substituting into the other, but it reveals the underlying algebraic structure of the curve.^[17] The arc length s of a parametric curve from t = a to t = b is given by the integral

s = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt,

assuming the curve is smooth and the integrand is continuous.^[18] This formula arises from approximating the curve with small line segments and taking the limit, measuring the total length along the path regardless of the parameterization speed.^[19] The tangent vector to the curve at a point corresponding to parameter t is (x'(t), y'(t)), which points in the direction of motion as t increases and its magnitude gives the speed along the curve.^[15] The curvature \kappa at that point, quantifying how sharply the curve bends, is

\kappa = \frac{\left| x'(t) y''(t) - y'(t) x''(t) \right|}{\left( (x'(t))^2 + (y'(t))^2 \right)^{3/2}},

derived from the rate of change of the unit tangent vector with respect to arc length.^[20] This expression holds for plane curves and assumes the parameterization is twice differentiable with non-zero speed.^[21] The orientation of the curve is determined by the direction in which points are traversed as t increases, providing a natural ordering along the path.^[22] Curves may be open, tracing a segment without closure, or closed, returning to the starting point after a finite parameter interval, which affects properties like enclosing areas or forming loops.^[5] Rational parametric curves arise when x(t) and y(t) are rational functions, typically ratios of polynomials, and are particularly significant for algebraic curves of genus zero, which admit such parametrizations.^[23] The degree of the parametrization refers to the maximum degree of the numerator and denominator polynomials, influencing the curve's complexity, while the genus measures the curve's topological "holes" and is zero for rational curves, enabling birational equivalence to the projective line.^[24]

Space Curves (3D)

In three-dimensional Euclidean space, a space curve is represented parametrically by a vector-valued function \vec{r}(t) = (x(t), y(t), z(t)), where t is the parameter, typically ranging over an interval, and x(t), y(t), and z(t) are smooth functions describing the coordinates along the curve.^[25] This form allows the curve to twist and turn freely in 3D, generalizing plane curves (where z(t) = 0) by incorporating depth and introducing torsion as a measure of out-of-plane twisting, which is absent in 2D.^[25] The direction of motion along the curve at any point is given by the unit tangent vector \vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}, where \vec{r}'(t) is the derivative with respect to t and |\vec{r}'(t)| is its magnitude, providing a normalized vector pointing along the instantaneous velocity.^[25] To analyze intrinsic geometric properties independent of the parameterization speed, curves are often reparameterized by arc length s, where s(t) = \int_a^t |\vec{r}'(u)| \, du, yielding \vec{r}(s) such that |\vec{r}'(s)| = 1.^[26] This arc-length parametrization simplifies computations of curvature and torsion by ensuring the parameter measures physical distance along the curve. For space curves, the Frenet-Serret frame provides an orthonormal basis adapted to the curve's local geometry, consisting of the unit tangent \vec{T}(s), the principal normal \vec{N}(s) (pointing toward the center of curvature), and the binormal \vec{B}(s) = \vec{T}(s) \times \vec{N}(s).^[27] The frame evolves according to the Frenet-Serret equations:

\begin{align*} \frac{d\vec{T}}{ds} &= \kappa(s) \vec{N}(s), \\ \frac{d\vec{N}}{ds} &= -\kappa(s) \vec{T}(s) + \tau(s) \vec{B}(s), \\ \frac{d\vec{B}}{ds} &= -\tau(s) \vec{N}(s), \end{align*}

where \kappa(s) is the curvature (rate of turning in the osculating plane) and \tau(s) is the torsion, defined as \tau(s) = -\frac{d\vec{B}}{ds} \cdot \vec{N}(s), quantifying the curve's deviation from planarity.^[27] A distinctive class of space curves exhibits constant curvature \kappa and constant torsion \tau, characteristic of uniform helices, which maintain a steady helical twist around an axis; by Lancret's theorem, such constant-ratio \kappa / \tau is necessary and sufficient for the curve to be a helix.^[28] In applications like computer graphics and path planning, space curves are often constructed via interpolation methods. Cubic Hermite interpolation defines a 3D path segment using endpoints \vec{P}_0, \vec{P}_1 and tangent vectors \vec{R}_0, \vec{R}_1 at those points, yielding the parametric form \vec{P}(t) = (2t^3 - 3t^2 + 1)\vec{P}_0 + (-2t^3 + 3t^2)\vec{P}_1 + (t^3 - 2t^2 + t)\vec{R}_0 + (t^3 - t^2)\vec{R}_1 for t \in [0,1], ensuring C^1 continuity when chaining segments by matching endpoints and tangents.^[29] Similarly, cubic Bézier interpolation uses four control points \vec{P}_0, \vec{P}_1, \vec{P}_2, \vec{P}_3, with the curve \vec{P}(t) = (1-t)^3 \vec{P}_0 + 3(1-t)^2 t \vec{P}_1 + 3(1-t) t^2 \vec{P}_2 + t^3 \vec{P}_3, interpolating the endpoints while the interior points guide the shape, and piecewise assembly achieves smooth 3D paths via C^0 or higher continuity at joins.^[29]

Generalizations to Higher Dimensions

Parametric equations generalize naturally to curves in n-dimensional Euclidean space \mathbb{R}^n, where a curve is represented by a vector-valued function \vec{r}(t) = (x_1(t), x_2(t), \dots, x_n(t)) with t ranging over an interval in \mathbb{R}.^[30] This parametrization allows the curve to trace paths through higher-dimensional spaces, such as in optimization or dynamical systems modeling. The tangent vector at a point is given by the derivative \vec{r}'(t), and for regularity, \vec{r}'(t) \neq \vec{0}. In computational contexts, the Jacobian matrix of \vec{r}(t), which for a single parameter is simply the row vector of partial derivatives, facilitates tangent space computations and sensitivity analysis.^[30] For more general structures, parametrized manifolds extend this idea to m-dimensional submanifolds embedded in \mathbb{R}^n (with m \leq n), defined by a smooth map \sigma: U \to \mathbb{R}^n where U \subset \mathbb{R}^m is open.^[31] The map \sigma must be regular, meaning its Jacobian matrix D\sigma(u) has full rank m at every u \in U, ensuring the image \sigma(U) forms an immersed submanifold with an m-dimensional tangent space spanned by the columns of D\sigma(u).^[31] This construction captures lower-dimensional objects in high-dimensional ambient spaces, such as graphs of functions or level sets. In differential geometry, the geometry of these parametrized manifolds is induced from the Euclidean metric on \mathbb{R}^n. The metric tensor g on the manifold, in local coordinates u^1, \dots, u^m, is given by the first fundamental form:

g_{ij} = \frac{\partial \sigma}{\partial u^i} \cdot \frac{\partial \sigma}{\partial u^j},

which measures lengths and angles via the dot product of partial derivatives.^[30] Christoffel symbols, which encode the Levi-Civita connection for covariant differentiation, are derived from the metric as

\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left( \frac{\partial g_{lj}}{\partial u^i} + \frac{\partial g_{il}}{\partial u^j} - \frac{\partial g_{ij}}{\partial u^l} \right),

enabling the study of geodesics and curvature without a coordinate-free formulation.^[30] These generalizations find applications in topology, where parametrizations describe compact manifolds like the n-sphere S^n, embedded in \mathbb{R}^{n+1} via hyperspherical coordinates: for the unit sphere,

x_1 = \cos \phi_1, \quad x_2 = \sin \phi_1 \cos \phi_2, \quad \dots, \quad x_n = \sin \phi_1 \cdots \sin \phi_{n-1} \cos \phi_n, \quad x_{n+1} = \sin \phi_1 \cdots \sin \phi_{n-1} \sin \phi_n,

with angles \phi_i \in [0, \pi] for i=1,\dots,n-1 and \phi_n \in [0, 2\pi).^[32] Similarly, the n-torus T^n = (S^1)^n is parametrized by n angular coordinates (\theta_1, \dots, \theta_n) \in [0, 2\pi)^n, embedding into \mathbb{R}^{2n} as (\cos \theta_1, \sin \theta_1, \dots, \cos \theta_n, \sin \theta_n), facilitating studies of fundamental groups and homology.^[33] Computationally, parametric equations in higher dimensions benefit from tensor representations, where the model is expressed via tensor products of univariate functions, enabling low-rank approximations for high-dimensional problems.^[34] For instance, a parametric map can be factorized into a correlation operator's spectral decomposition, yielding sparse tensors that reduce storage and computation costs in simulations, as seen in uncertainty quantification for PDEs.^[34] This approach unifies linear and affine parametrizations, supporting efficient numerical methods in programming environments like MATLAB or Python libraries for tensor algebra.^[34]

Specific Examples of Parametric Curves

Conic Sections

Conic sections, which include parabolas, ellipses, hyperbolas, and circles, can be effectively parametrized using trigonometric, hyperbolic, or polynomial functions, allowing for the representation of their algebraic properties in terms of a parameter t. These parametrizations facilitate the study of their geometric features, such as foci, directrices, and asymptotes, while enabling elimination of the parameter to recover the standard implicit quadratic equation.^[35] The parabola is defined as the set of points equidistant from a fixed point (focus) and a fixed line (directrix). A standard parametric representation for the parabola with focus at (a, 0) and directrix x = -a is given by

x = at^2, \quad y = 2at,

where a > 0 and t \in (-\infty, \infty). This form arises from the focus-directrix property, where the parameter t represents the slope of the tangent to the parabola at that point. Eliminating t yields the implicit equation y^2 = 4ax, highlighting the quadratic nature of the curve.^[36]^[37]^[38] For the ellipse, characterized by two foci and an eccentricity e < 1, where e = c/a with c the distance from center to focus and a the semi-major axis, the standard parametric equations are

x = a \cos t, \quad y = b \sin t,

with b = a \sqrt{1 - e^2} the semi-minor axis and t \in [0, 2\pi). These trigonometric parametrizations leverage the identity \cos^2 t + \sin^2 t = 1 to satisfy the implicit form \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, providing a uniform traversal of the closed curve. The eccentricity quantifies the ellipse's deviation from a circle, with e = 0 corresponding to the circular case.^[5]^[39] The circle emerges as a special ellipse when a = b = r, yielding the parametrization

x = r \cos t, \quad y = r \sin t, \quad t \in [0, 2\pi).

This traces the circle x^2 + y^2 = r^2 centered at the origin with radius r. An alternative rational parametrization, derived from stereographic projection from the north pole (0, r) onto the line y = -r, is

x = \frac{2rt}{1 + t^2}, \quad y = r \frac{1 - t^2}{1 + t^2}, \quad t \in \mathbb{R},

which covers the entire circle except possibly one point and uses rational functions, useful in algebraic geometry for avoiding trigonometric functions.^[5]^[40] The hyperbola, with eccentricity e > 1 and two branches separated by asymptotes, admits parametric forms using either secant-tangent or hyperbolic functions. For the right branch of \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, where b = a \sqrt{e^2 - 1}, one representation is

x = a \sec t, \quad y = b \tan t, \quad t \in (-\pi/2, \pi/2),

exploiting \sec^2 t - \tan^2 t = 1. Alternatively,

x = a \cosh t, \quad y = b \sinh t, \quad t \in (-\infty, \infty),

utilizes hyperbolic identities \cosh^2 t - \sinh^2 t = 1, offering a parametrization without discontinuities. The asymptotes are lines y = \pm (b/a) x, approached as t \to \pm \infty in either form. The left branch follows by negating x.^[41]^[42]^[39] In general, any conic section is represented implicitly by the quadratic equation

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,

where the discriminant B^2 - 4AC distinguishes the type: negative for ellipses (including circles), zero for parabolas, and positive for hyperbolas. Parametric equations for a general conic can be derived via projective transformations from standard forms, and eliminating the parameter t from such representations recovers this quadratic form, confirming the algebraic degree two classification.^[43]^[8]

Cycloids and Trochoids

A cycloid is the curve traced by a point on the rim of a circle of radius a as the circle rolls along a straight line without slipping. The parametric equations for a cycloid with cusps oriented upward are given by

\begin{align*} x &= a(t - \sin t), \\ y &= a(1 - \cos t), \end{align*}

where t is the parameter representing the angle of rotation in radians.^[44] Each arch of the cycloid spans from one cusp to the next, corresponding to t increasing by $2\pi, with the curve exhibiting cusps at points where t = 2k\pi for integer k, and reaching a maximum height of $2a midway between cusps.^[44] The parametric speed along the cycloid, or the magnitude of the velocity vector, is v(t) = 2a \left| \sin(t/2) \right|, which vanishes at the cusps and achieves a maximum of $2a at the peaks of the arches./12%3A_Parametric_Equations/12.01%3A_Parametric_Equations) The evolute of a cycloid is another cycloid, congruent and translated vertically by $2a from the original, a property that distinguishes it among plane curves.^[44] Trochoids generalize cycloids to curves generated by a point at a fixed distance c from the center of a rolling circle of radius b along a fixed circle of radius a. When the rolling circle moves externally around the fixed circle and c = b, the resulting epicycloid has parametric equations

\begin{align*} x &= (a + b) \cos t - b \cos\left( \frac{a + b}{b} t \right), \\ y &= (a + b) \sin t - b \sin\left( \frac{a + b}{b} t \right). \end{align*}

The number of cusps in an epicycloid depends on the ratio a/b, producing cardiods when a = b and nephroids when a = 2b.^[45] For internal rolling with c = b, the hypocycloid follows similar forms but with the ratio (a - b)/b, yielding astroids for a = 4b and deltoids for a = 3b.^[46] Hypotrochoids extend this to internal rolling where c \neq b, with parametric equations

\begin{align*} x &= (a - b) \cos t + c \cos\left( \frac{a - b}{b} t \right), \\ y &= (a - b) \sin t - c \sin\left( \frac{a - b}{b} t \right). \end{align*}

A special case is the deltoid (or tricuspoid), obtained when a = 3b and c = b, forming a three-cusped hypocycloid used in mechanisms like the Spirograph toy.^[47] The curve's shape varies with the ratio a/b and c/b; for rational ratios, it closes after a finite number of rotations, producing star-like patterns with the number of cusps or loops determined by the denominator of the reduced ratio.^[47] In the brachistochrone problem, which seeks the curve of fastest descent for a particle sliding under gravity between two points, the solution is an inverted cycloid arc, as independently discovered by Johann Bernoulli, Leibniz, L'Hôpital, and Newton in 1696–1697.^[48] This property arises because the cycloid minimizes the descent time compared to other paths like straight lines or parabolas, with the optimal radius a scaled to fit the endpoints.^[48]

Lissajous and Other Periodic Curves

Lissajous curves arise from the superposition of two perpendicular harmonic oscillations with potentially different amplitudes, frequencies, and phase shifts, producing a wide variety of closed or open patterns depending on these parameters. The parametric equations for a Lissajous curve in the plane are given by

\begin{align*} x(t) &= A \sin(\omega t + \phi), \\ y(t) &= B \sin(\nu t), \end{align*}

where A and B are the amplitudes, \omega and \nu are the angular frequencies, and \phi is the phase difference between the oscillations.^[49] When the frequency ratio \omega / \nu is rational, such as 1:1, the curve closes to form an ellipse (or a circle if A = B and \phi = \pm \pi/2); for ratios like 1:2 or 3:2, it generates more complex figures-of-eight or parabolic-like loops.^[49] These curves were originally studied in the context of coupled oscillators, where they visualize the resulting trajectories.^[50] The shape of a Lissajous curve is highly sensitive to the phase difference \phi, which can transform the pattern from a straight line (\phi = 0 or \pi) to tilted ellipses or bow-like forms (e.g., \phi = \pi/4 yielding asymmetric loops resembling bows in certain frequency ratios).^[51] In physical systems like coupled pendulums or electronic oscillators, these curves exhibit stability when driven at resonant frequencies, maintaining closed paths that reflect the system's harmonic balance without diverging into aperiodic fills.^[50] Such properties make Lissajous figures valuable for analyzing phase relationships and frequency ratios in oscillatory experiments.^[49] Knot curves, such as the trefoil knot, can be represented parametrically and viewed as closed paths in a 2D projection, illustrating intertwined periodic motions. A standard trigonometric parametrization for the trefoil knot in 3D is

\begin{align*} x(t) &= \cos(2t) \left(3 + \cos(3t)\right), \\ y(t) &= \sin(2t) \left(3 + \cos(3t)\right), \\ z(t) &= \sin(3t), \end{align*}

with t \in [0, 2\pi]; projecting this onto a plane (e.g., the xy-plane) yields a 2D curve that traces a three-lobed pattern with three self-intersections characteristic of the knot.^[52] This projection highlights the periodic, self-intersecting nature of the curve while preserving its topological knotting in the visual representation.^[53] More general closed periodic curves can be parametrized using Fourier series, expressing the coordinates as sums of sines and cosines to approximate smooth, looping paths. For a plane curve, the parametrization takes the form

\begin{align*} x(t) &= a_0 + \sum_{k=1}^n \left( a_k \cos(kt) + b_k \sin(kt) \right), \\ y(t) &= c_0 + \sum_{k=1}^n \left( c_k \cos(kt) + d_k \sin(kt) \right), \end{align*}

where the coefficients a_k, b_k, c_k, d_k are determined by Fourier analysis of the curve's boundary, ensuring closure for periodic t over [0, 2\pi].^[54] This approach, known as Fourier descriptors, captures the curve's shape invariants under translation, rotation, and scaling, making it useful for representing complex periodic contours like those in boundary detection.^[54] Higher-order terms allow approximation of arbitrary closed curves, with rational frequency components ensuring the path remains bounded and non-self-overlapping in simple cases.^[55] Viviani's curve provides another example of a periodic space curve defined as the intersection of a sphere of radius $2a centered at the origin and a cylinder of radius a tangent to the sphere along the line x = a, y = 0. Its parametric equations are

\begin{align*} x(t) &= a \left(1 + \cos t\right), \\ y(t) &= a \sin t, \\ z(t) &= 2a \sin\left(\frac{t}{2}\right), \end{align*}

for t \in [0, 4\pi], tracing a figure-eight shape that lies on both surfaces and closes after two periods due to the half-angle in the z-coordinate.^[56] This curve demonstrates how parametric forms can elegantly capture intersections of quadrics, with its periodic nature arising from the trigonometric functions matching the cylindrical and spherical geometries.^[56] In 3D extensions, such periodic curves can project to 2D Lissajous-like patterns while retaining spatial loops akin to knots.^[56]

Parametric Surfaces and Solids

Definition and Parametrization

A parametric surface in three-dimensional Euclidean space is defined as the image of a continuously differentiable mapping from a two-dimensional domain D \subset \mathbb{R}^2 to \mathbb{R}^3, extending the concept of parametric curves that use a single parameter to trace one-dimensional paths.^[57] Specifically, it is given by a vector-valued function \vec{r}(u,v) = (x(u,v), y(u,v), z(u,v)), where u and v are parameters ranging over the domain D, which is typically an open or closed region such as a rectangle, disk, or annulus.^[58] This parametrization allows the surface to be locally represented as a graph over the uv-plane, with the mapping \vec{r} being smooth and injective on D to avoid self-intersections, though global injectivity is not always required.^[59] The geometry of the parametric surface is described by its first fundamental form, which captures the intrinsic metric properties independent of the embedding in \mathbb{R}^3. This form is expressed as ds^2 = E\, du^2 + 2F\, du\, dv + G\, dv^2, where E = \vec{r}_u \cdot \vec{r}_u, F = \vec{r}_u \cdot \vec{r}_v, and G = \vec{r}_v \cdot \vec{r}_v, with \vec{r}_u = \frac{\partial \vec{r}}{\partial u} and \vec{r}_v = \frac{\partial \vec{r}}{\partial v} denoting the partial derivatives.^[60] The coefficients E, F, and G determine lengths, angles, and areas on the surface, forming a Riemannian metric that measures distances along the surface.^[61] A key extrinsic feature is the unit normal vector \vec{N} = \frac{\vec{r}_u \times \vec{r}_v}{|\vec{r}_u \times \vec{r}_v|}, which is well-defined provided the cross product is nonzero (ensuring the parametrization is regular).^[57] The Gaussian curvature K, an intrinsic invariant, is then given by K = \frac{LN - M^2}{EG - F^2}, where L, M, and N are coefficients of the second fundamental form involving second partial derivatives and the normal.^[62] Parametric surfaces can be classified by orientability, which depends on whether a consistent choice of normal vector field exists across the entire surface. An orientable surface admits a continuous, nowhere-vanishing normal \vec{N} that does not reverse direction, allowing a two-sided distinction, whereas non-orientable surfaces like the Möbius strip do not.^[63] Regarding boundaries, surfaces are closed if the domain D is compact without boundary (e.g., a torus), yielding a boundaryless manifold, or open if D has a boundary, resulting in a surface with edges where the parametrization restricts to curves.^[64] A special type of parametrization is the isothermal one, which is conformal and preserves angles between curves on the surface, satisfying E = G and F = 0 in the first fundamental form, thus mapping the metric to a scaled Euclidean metric in the parameter domain.^[65] Such parametrizations exist locally for any surface and are useful for uniform sampling and visualization.^[66]

Ruled Surfaces and Developables

A ruled surface is a surface that can be generated by the motion of a straight line in space, known as a ruling, along a curve. In parametric form, it is expressed as \vec{r}(u,v) = \vec{a}(u) + v \vec{b}(u), where \vec{a}(u) traces a directrix curve and \vec{b}(u) provides the direction of the rulings varying with u, ensuring that for fixed u, the line segment varies linearly with v.^[67] This parametrization highlights the linear structure along the v-direction, distinguishing ruled surfaces from more general parametric surfaces.^[67] Developable surfaces form a special class of ruled surfaces characterized by zero Gaussian curvature, meaning they possess a single tangent plane along each ruling and can be isometrically mapped onto a plane without distortion, such as stretching or tearing.^[67] This property arises because the surface is intrinsically flat in one direction, allowing it to be "unrolled" like paper.^[67] Classic examples include cylinders, where rulings are parallel straight lines, and cones, where rulings converge to a vertex; both maintain constant tangent planes along their generators.^[67] Notable non-developable ruled surfaces include the hyperboloid of one sheet, which is doubly ruled—meaning it admits two distinct families of rulings covering the surface. Its parametric equations for one family are given by

\begin{align*} x(u,v) &= a (\cosh u \cos v), \\ y(u,v) &= a (\cosh u \sin v), \\ z(u,v) &= c \sinh u, \end{align*}

with the second family obtained by interchanging hyperbolic functions, demonstrating the surface's linear generators.^[68] Another key example is the tangent developable, constructed from a space curve \vec{d}(s) parametrized by arc length s, as \vec{r}(s, \beta) = \vec{d}(s) + \beta \vec{d}'(s), where the rulings are the tangent lines to the curve.^[69] These surfaces illustrate how ruled parametrizations can generate complex geometries while preserving linearity in one parameter. Key properties of ruled and developable surfaces include the fact that rulings on developables serve as geodesics, being the straightest paths on the surface due to the zero geodesic curvature in the ruling direction.^[67] The Dupin indicatrix provides a local classification of points: for developables, it yields parabolic points where one principal curvature vanishes (Gaussian curvature K=0), manifesting as parallel lines in the indicatrix, in contrast to elliptic (K>0) or hyperbolic (K<0) points on other surfaces.^[61] This classification underscores the flattenability of developables, as the indicatrix's form aligns with unbendable planes.^[61] Parametrizing ruled surfaces presents challenges, particularly with singularities that occur along edges where the tangent plane becomes undefined or the parametrization degenerates, such as at cuspidal edges in tangent developables.^[70] These arise when the derivative of the geodesic curvature vanishes, leading to fixed lines or higher-order singularities like swallowtails, complicating numerical stability and requiring careful choice of directrix to avoid regression curves.^[70] In non-developable cases, such singularities highlight the surface's saddle-like behavior without the global flattenability of developables.^[70]

Quadric Surfaces

Quadric surfaces are algebraic surfaces of degree two defined by quadratic equations in three variables, generalizing conic sections to three dimensions within algebraic geometry.^[71] Their parametric representations facilitate visualization, computation, and applications in computer-aided geometric design by expressing points on the surface using two parameters, often adapted from spherical or hyperbolic coordinates. These parametrizations reveal the surfaces' topological and geometric properties, such as compactness for ellipsoids or hyperbolicity for hyperboloids.^[72] The ellipsoid, a bounded quadric resembling a stretched sphere, admits a parametric form derived from spherical coordinates scaled by semi-axes lengths a, b, and c:

\begin{align*} x &= a \cos \theta \cos \phi, \\ y &= b \cos \theta \sin \phi, \\ z &= c \sin \theta, \end{align*}

where \theta \in [0, \pi] and \phi \in [0, 2\pi). This representation covers the entire surface without singularities except at the poles.^[72] Hyperboloids, unbounded quadrics with hyperbolic cross-sections, have distinct forms for one and two sheets. The hyperboloid of one sheet, connected and ruled, uses hyperbolic functions for its parametrization:

\begin{align*} x &= a \cosh u \cos v, \\ y &= b \cosh u \sin v, \\ z &= c \sinh u, \end{align*}

with u \in \mathbb{R} and v \in [0, 2\pi), yielding the implicit equation \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1. An alternative trigonometric form employs secant and tangent: x = a \sec \theta \cos \phi, y = b \sec \theta \sin \phi, z = c \tan \theta, for \theta \in (-\pi/2, \pi/2) and \phi \in [0, 2\pi).^[73] The hyperboloid of two sheets, disconnected with two separate components, is parametrized as:

\begin{align*} x &= a \sinh u \cos v, \\ y &= b \sinh u \sin v, \\ z &= c \cosh u, \end{align*}

for the upper sheet with u \geq 0 and z = c \cosh u, or for the lower sheet with u \geq 0 and z = -c \cosh u, and v \in [0, 2\pi), satisfying \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1. These forms highlight the surface's separation along the axis.^[73]^[74] Paraboloids are quadratic in two variables and linear in the third, producing open surfaces. The elliptic paraboloid, bowl-shaped, has the parametric equations:

\begin{align*} x &= a s \cos \theta, \\ y &= b s \sin \theta, \\ z &= s^2, \end{align*}

where s \geq 0 and \theta \in [0, 2\pi), corresponding to \frac{x^2}{a^2} + \frac{y^2}{b^2} = z. The hyperbolic paraboloid, saddle-shaped, uses bilinear parameters:

\begin{align*} x &= a u, \\ y &= b v, \\ z &= u^2 - v^2, \end{align*}

with u, v \in \mathbb{R}, for the equation \frac{x^2}{a^2} - \frac{y^2}{b^2} = z. These parametrizations emphasize the elliptic or hyperbolic nature of horizontal traces.^[75] For a general quadric surface given by A x^2 + B y^2 + C z^2 + D x y + E x z + F y z + G x + H y + I z + J = 0, parametrization involves diagonalizing the associated symmetric matrix via its eigenvalues and eigenvectors to align with principal axes, reducing to a standard form like an ellipsoid or hyperboloid, then applying the appropriate parametric map. Alternatively, rational quadratic parametrizations can be obtained via stereographic projection from a nonsingular point on the surface to a parameter plane, yielding birational maps suitable for computational geometry.^[73]^[72] Singular quadrics, where the defining matrix has rank less than 4, include cones and cylinders as limiting cases of nonsingular quadrics. A cone, irreducible and rank-3, is parametrized linearly as:

\begin{align*} x &= a u \cos v, \\ y &= b u \sin v, \\ z &= c u, \end{align*}

with u \in \mathbb{R} and v \in [0, 2\pi), satisfying \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0, emerging as the limit of a hyperboloid when the constant term approaches zero. Cylinders, rank-2 and extrusions of conics, have the form:

\begin{align*} x &= a \cos u, \\ y &= b \sin u, \\ z &= v, \end{align*}

for u \in [0, 2\pi) and v \in \mathbb{R}, as in \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, obtained as a limiting degenerate ellipsoid or hyperboloid with infinite extent in one direction.^[72]

Parametric Solids

Parametric solids extend the concept to three-dimensional volumes, defined as the image of a continuously differentiable mapping from a three-dimensional domain D \subset \mathbb{R}^3 to \mathbb{R}^3, given by \vec{r}(u,v,w) = (x(u,v,w), y(u,v,w), z(u,v,w)), where u, v, w are parameters. This allows representation of solid objects like spheres or polyhedra. For example, the unit ball can be parametrized using spherical coordinates: x = \rho \sin \phi \cos \theta, y = \rho \sin \phi \sin \theta, z = \rho \cos \phi, with \rho \in [0,1], \phi \in [0, \pi], \theta \in [0, 2\pi). Such parametrizations are useful in computer graphics for rendering volumes and in physics for modeling material distributions.^[76]

Conversions Between Representations

Implicitization Process

The implicitization process converts a parametric representation of a curve or surface, given by equations such as x = f(t), y = g(t), into an implicit equation F(x, y, \dots) = 0 that defines the same geometric object without the parameter t. This elimination is crucial in algebraic geometry and computer-aided design for tasks like intersection computations and membership testing.^[14] For rational parametric curves, where x = \frac{p(t)}{r(t)} and y = \frac{q(t)}{s(t)} with polynomials p, q, r, s, implicitization often employs the Sylvester resultant after clearing denominators to form polynomials in t. The resultant with respect to t yields a polynomial in x and y whose roots describe the curve. For instance, consider the rational quadratic curve parametrized by x = t, y = t^2; clearing any denominators (none here) gives the polynomials x - t = 0 and y - t^2 = 0, and their Sylvester resultant is y - x^2 = 0, the implicit equation of a parabola. In general, for a properly parametrized rational curve of degree d, the implicit equation has degree d.^[14]^[77] Gröbner bases provide a computational framework for implicitization in multivariate rational cases, generating an ideal from the parametric equations and eliminating parameters via a suitable monomial order. Specifically, for equations x_i h(t) - f_i(t) = 0 where h, f_i are polynomials, the Gröbner basis of the ideal allows extraction of the elimination ideal in the coordinate ring, yielding the implicit prime ideal. This method handles improper parametrizations and ensures completeness by computing a basis for the implicit variety. Algorithms based on this approach solve for the implicit form and detect properness of the parametrization.^[78] For parametric lines in three dimensions, given by \vec{r}(t) = \vec{p} + t \vec{d}, implicitization uses the cross product to enforce collinearity: (\vec{r} - \vec{p}) \times \vec{d} = \vec{0}, which expands to two independent linear equations in the coordinates of \vec{r}. This represents the line as the solution set to a system of two planes intersecting along the line. Challenges in implicitization include degree inflation, where an improper parametrization leads to extraneous factors in the resultant, increasing the apparent degree of the implicit equation beyond the geometric degree. Additionally, the process can introduce singularities or extra components if base points exist in the parametrization, complicating the variety's structure. For low-degree cases like quadrics, the Dixon resultant offers an efficient matrix-based elimination, forming a determinant that implicitizes the surface without base point issues when properly constructed.^[14]^[79]^[80] Symbolic computation tools, such as those implementing resultants and Gröbner bases, facilitate practical implicitization for higher-degree cases, though computational cost grows with degree.^[78]

Conversion to Explicit Forms

Converting parametric equations x = x(t) and y = y(t) to an explicit form y = f(x) involves solving for the parameter t in terms of x and substituting into the equation for y.^[81] This process requires inverting the function x(t) to obtain t = x^{-1}(x), followed by y = y(x^{-1}(x)).^[81] For this inversion to yield a single-valued explicit function, x(t) must be strictly monotonic (either increasing or decreasing) over the domain of interest, ensuring a one-to-one correspondence between t and x. However, many parametric curves cannot be represented as a single explicit function y = f(x) due to multi-valued branches or vertical tangents. For instance, a circle parametrized by x = \cos t, y = \sin t cannot be expressed as a single explicit equation because it requires two branches, such as y = \pm \sqrt{1 - x^2}, to cover the full curve.^[4] This limitation highlights why parametric forms are often preferred for closed or non-monotonic curves, as they avoid such discontinuities.^[4] Obtaining an explicit expression for arc length as a function of x, s(x), from parametric equations is generally challenging and rarely yields a closed-form solution, necessitating numerical integration in most cases.^[82] The arc length formula in parametric form is s = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt, but converting to an explicit integral with respect to x requires \frac{dy}{dx} and often leads to elliptic integrals or other non-elementary functions.^[83] For straight lines, however, the explicit arc length simplifies via the Pythagorean theorem to s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.^[83] In polar coordinates, a curve given parametrically as r = r(\theta), with x = r(\theta) \cos \theta and y = r(\theta) \sin \theta, can sometimes be converted to an explicit Cartesian form, but deriving r explicitly in terms of x and y typically results in an implicit relation like r = \sqrt{x^2 + y^2}.^[84] For spline curves, such as those defined by parametric B-splines, conversion to explicit form involves evaluating the parametric polynomials over each segment to obtain piecewise explicit polynomial expressions y = f_i(x) for intervals [x_i, x_{i+1}].^[85] This piecewise nature allows for accurate representation of complex curves while maintaining computational efficiency, though inversion per segment requires ensuring local monotonicity in x(t).^[86]

Rational vs. Non-Rational Parametrizations

A rational parametrization of a curve expresses the coordinates as ratios of polynomials in a parameter t, typically over a field such as the rationals or reals.^[87] This form provides a birational map from the projective line to the curve, covering all but finitely many points.^[24] For example, the unit circle x^2 + y^2 = 1 admits the rational parametrization

x = \frac{1 - t^2}{1 + t^2}, \quad y = \frac{2t}{1 + t^2},

which traces the curve exactly as t varies over the reals, excluding the point at infinity corresponding to t = \infty.^[88] In contrast, non-rational parametrizations employ transcendental functions, such as trigonometric or exponential ones, which cannot be expressed as ratios of polynomials.^[89] A classic example is the circular helix, given by

x = \cos t, \quad y = \sin t, \quad z = t,

where the involvement of sine and cosine precludes a rational form, as the helix is a transcendental curve not algebraic over the rationals.^[90] Such parametrizations are essential for curves lacking algebraic structure, but they introduce approximations or infinite series when numerical evaluation is required.^[87] For algebraic curves, the existence of a rational parametrization is tied to the curve's genus: only irreducible curves of genus zero admit such a parametrization, establishing birational equivalence to the projective line \mathbb{P}^1.^[87] Curves of higher genus, like smooth cubics (genus one), cannot be rationally parametrized and require elliptic or other non-rational representations for traversal.^[24] The degree of a rational parametrization is minimal when it is proper, matching the degree of the curve for birational maps; for instance, conics require degree two, while lines use degree one.^[87] Rational parametrizations offer key advantages in computational contexts, such as computer-aided design, by enabling exact arithmetic operations without floating-point errors inherent in transcendental evaluations.^[89] In number theory, the rational parametrization of the circle generates Pythagorean triples via integer substitutions for t, yielding solutions to a^2 + b^2 = c^2 like (3,4,5) from t = 1.^[88] This exactness facilitates implicitization in conversions between parametric and implicit forms, preserving algebraic structure.^[87]

Applications

Kinematics and Physics

In kinematics and physics, parametric equations provide a powerful framework for describing the trajectories and dynamics of particles and systems as functions of time or other parameters, enabling the analysis of position, velocity, and acceleration in a unified manner.^[4] The position of a particle is typically represented as a vector function \vec{r}(t), where t is the parameter, often time; the velocity is then \vec{v}(t) = \frac{d\vec{r}}{dt}, and the acceleration is \vec{a}(t) = \frac{d^2\vec{r}}{dt^2}.^[91] This parametrization allows for the modeling of curved paths in multiple dimensions, facilitating the application of Newton's laws or Lagrangian mechanics to compute forces and energies along the trajectory.^[81] A classic example is projectile motion under constant gravity, where the horizontal and vertical components are decoupled. The parametric equations are x(t) = (v_0 \cos \theta) t and y(t) = (v_0 \sin \theta) t - \frac{1}{2} g t^2, with initial speed v_0, launch angle \theta, and gravitational acceleration g.^[92] The range, or horizontal distance traveled upon return to the initial height, is R = \frac{v_0^2 \sin 2\theta}{g}, maximized at \theta = 45^\circ, while the maximum height is H = \frac{(v_0 \sin \theta)^2}{2g}.^[93] These expressions derive from setting the vertical velocity to zero at the peak and solving for the time of flight, highlighting how parametric forms simplify the elimination of t to obtain the trajectory equation y = x \tan \theta - \frac{g x^2}{2 v_0^2 \cos^2 \theta}.^[94] For oscillatory systems, parametric equations capture periodic motion, as in the simple harmonic oscillator where displacement is x(t) = A \cos(\omega t + \phi), with amplitude A, angular frequency \omega = \sqrt{k/m}, and phase \phi.^[95] The velocity is v(t) = -A \omega \sin(\omega t + \phi), and acceleration a(t) = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t), restoring the system toward equilibrium. Energy conservation is evident, as total mechanical energy E = \frac{1}{2} k A^2 remains constant, partitioning between kinetic \frac{1}{2} m v^2 and potential \frac{1}{2} k x^2 forms without dissipation in the ideal case.^[96] In rigid body dynamics, screw theory parametrizes general spatial motions using helical axes, as per Chasles' theorem, which states that any rigid body displacement is equivalent to a rotation about an axis combined with a translation along the same axis (a screw motion).^[97] The motion can be expressed via exponential coordinates on the Lie algebra of the special Euclidean group SE(3), where a twist \xi generates the displacement g = e^{\hat{\xi} \theta} for angle \theta, encapsulating both rotational and translational components along the helical axis.^[98] This unification simplifies analysis in robotics and mechanics, reducing six-dimensional rigid body motions to parameters along screws. In special relativity, parametric equations describe timelike worldlines in Minkowski spacetime, parametrized by proper time \tau to reflect the invariant interval along the path. The four-position x^\mu(\tau) satisfies ds^2 = -c^2 d\tau^2, ensuring the proper time is the time measured by a comoving clock, maximizing \tau for inertial paths between events.^[99] The four-velocity u^\mu = \frac{dx^\mu}{d\tau} is normalized such that u^\mu u_\mu = -c^2, facilitating Lorentz-invariant formulations of particle dynamics and geodesic motion in curved spacetimes.^[100]

Computer-Aided Design and Graphics

In computer-aided design (CAD), parametric equations enable precise modeling of complex curves and surfaces through control points and basis functions, facilitating interactive shape manipulation and manufacturing. Bézier curves, a foundational parametric representation, are defined using Bernstein polynomials of degree n as \mathbf{B}(t) = \sum_{i=0}^{n} \binom{n}{i} t^i (1-t)^{n-i} \mathbf{P}_i, where \mathbf{P}_i are control points forming a control polygon that approximates the curve while ensuring endpoint interpolation and convexity preservation.^[101] These curves, developed for automotive design at Renault, allow designers to iteratively adjust shapes via control points without recalculating the entire geometry, supporting applications in free-form surface modeling.^[101] Non-uniform rational B-splines (NURBS) extend Bézier curves to higher degrees and rational forms, providing a unified framework for both free-form and exact conic representations in CAD systems. A NURBS curve is given by \mathbf{C}(u) = \frac{\sum_{i=0}^{n} w_i \mathbf{P}_i N_{i,p}(u)}{\sum_{i=0}^{n} w_i N_{i,p}(u)}, where N_{i,p}(u) are B-spline basis functions defined by a non-uniform knot vector, and weights w_i enable representation of conics like circles via the conic shape factor, with w_1 = \cos(\theta/2) for a circular arc of angle \theta.^[102] Adopted in early systems like Boeing's TIGER (1979) and SDRC's GEOMOD (1983), NURBS ensure affine invariance, local control, and efficient evaluation, making them standard for precise 3D modeling in industries such as aerospace and automotive.^[102] In computer graphics and animation, parametric equations generate smooth paths between keyframes, with Catmull-Rom splines offering C^1 continuity and interpolation through control points \mathbf{P}_{i-1}, \mathbf{P}_i, \mathbf{P}_{i+1}, \mathbf{P}_{i+2} via the hermite form \mathbf{S}(t) = (2t^3 - 3t^2 + 1) \mathbf{P}_i + (t^3 - 2t^2 + t) \Delta \mathbf{P}_i + (-2t^3 + 3t^2) \mathbf{P}_{i+1} + (t^3 - t^2) \Delta \mathbf{P}_{i+1}, where \Delta \mathbf{P}_i = \mathbf{P}_{i+1} - \mathbf{P}_i. This parameterization ensures tension-free motion for keyframe interpolation in tools like Autodesk Maya, preserving velocity at knots for natural object trajectories without overshooting. Parametric representations are essential in ray tracing for efficient intersection computations, where rays are parameterized as \mathbf{R}(u) = \mathbf{O} + u \mathbf{D} and intersected with surfaces \mathbf{S}(s,t) by solving \mathbf{S}(s,t) - \mathbf{R}(u) = \mathbf{0} using multivariate Newton iteration: \mathbf{v}_{k+1} = \mathbf{v}_k - J_f(\mathbf{v}_k)^{-1} f(\mathbf{v}_k), with \mathbf{v} = (s,t,u) and Jacobian J_f.^[103] Interval analysis bounds the parameter space to guarantee convergence, enabling robust tests against polynomial patches in rendering pipelines like those in Pixar’s RenderMan.^[103] Modern graphics leverage GPU shaders for real-time parametric deformations, implementing vertex programs that apply functions like wave distortions f(x,y,z) = (x, A \sin(\omega \sqrt{x^2 + z^2} + \phi), z) directly to coordinates, with normals transformed via the Jacobian matrix's inverse transpose to maintain lighting consistency.^[104] This approach, requiring differentiable parameterizations, achieves high performance on parallel hardware without neighbor dependencies, supporting interactive editing in tools like Unity and Unreal Engine.^[104]

Integer and Diophantine Equations

Parametric equations play a crucial role in solving Diophantine equations, which seek integer solutions to polynomial equations, by providing systematic ways to generate infinite families of solutions from a finite set of parameters. In number theory, these parametrizations often exploit algebraic structures like units in quadratic fields or continued fraction expansions to enumerate all or primitive solutions, bridging geometry and arithmetic. For Pythagorean triples, which satisfy x^2 + y^2 = z^2 with positive integers x, y, z, primitive solutions (where \gcd(x,y,z)=1) are generated parametrically by integers m > n > 0 with m and n coprime and not both odd: x = m^2 - n^2, y = 2mn, z = m^2 + n^2. This form, attributed to Euclid, produces all primitive triples, and general solutions scale by a positive integer k. Pell equations of the form x^2 - d y^2 = 1, where d > 0 is a square-free integer, admit parametric solutions derived from the continued fraction expansion of \sqrt{d}. The fundamental solution (x_1, y_1) generates all others via x_k + y_k \sqrt{d} = (x_1 + y_1 \sqrt{d})^k for positive integers k, yielding infinitely many solutions in a recurrent parametric sequence. Linear Diophantine equations ax + by = c, with integers a, b, c and \gcd(a,b)=d dividing c, have parametric integer solutions once a particular solution (x_0, y_0) is found: x = x_0 + (b/d)t, y = y_0 - (a/d)t for any integer parameter t. This parametrization captures the general solution lattice in the plane. In the context of Fermat's Last Theorem, which asserts no positive integer solutions to x^n + y^n = z^n for n > 2, no simple parametric forms exist for n > 2, unlike the quadratic case; however, for cubic equations like x^3 + y^3 = z^3 + k, certain parametric families have been identified for small k, though they do not contradict the theorem. Generating functions for integer sequences, such as partitions, can be parametrized to solve Diophantine relations; for instance, the generating function \sum p(n) q^n = \prod_{k=1}^\infty (1 - q^k)^{-1} leads to parametric recurrences that enumerate solutions to partition equations like the number of ways to sum to n.

Linear Systems and Optimization

In linear algebra, the general solution to a system of linear equations Ax = b, where A is an m \times n matrix with m \leq n, is often expressed in parametric form when the system is consistent and underdetermined, featuring free variables corresponding to non-pivot columns in the row echelon form of A. This representation explicitly describes all solutions as \mathbf{x} = \mathbf{x}_p + \sum_{i=1}^k t_i \mathbf{v}_i, where \mathbf{x}_p is a particular solution, the t_i are free parameters (typically real numbers), and the \mathbf{v}_i form a basis for the null space of A.^[105]^[106] The process begins by augmenting A with \mathbf{b} and performing Gaussian elimination to obtain the reduced row echelon form, identifying pivot and free variables. For instance, consider the system

\begin{cases} 2x + y + 12z = 1, \\ x + 2y + 9z = -1. \end{cases}

Row reduction yields x = 1 - 5z, y = -1 - 2z, with z free, so the parametric equations are x = 1 - 5t, y = -1 - 2t, z = t for parameter t \in \mathbb{R}, tracing a line in \mathbb{R}^3. This form highlights the geometric interpretation: the solution set is an affine subspace, such as a line or plane, parametrized affinely.^[105]^[106] Such parametrizations are essential for understanding the dimension and structure of solution spaces; the number of free variables equals the nullity of A, determining whether solutions are unique (nullity 0), infinite (nullity > 0), or nonexistent (inconsistent system). In applications like computer graphics or robotics, these parametric lines or planes model trajectories or constraints efficiently.^[105] In optimization, parametric equations arise prominently in parametric linear programming (PLP), where problem data such as the right-hand side \mathbf{b}(t) or objective coefficients \mathbf{c}(t) depend linearly on a parameter t, and the optimal solution \mathbf{x}^*(t) is analyzed as a function of t. Seminal work established that for a PLP \min \{\mathbf{c}(t)^T \mathbf{x} \mid A\mathbf{x} = \mathbf{b}(t), \mathbf{x} \geq \mathbf{0} \}, the optimal basis remains constant over intervals of t, yielding affine parametric representations \mathbf{x}^*(t) = B^{-1} \mathbf{b}(t) within each critical region, where B is the basis matrix. This piecewise affine structure allows the optimal value function v(t) = \mathbf{c}(t)^T \mathbf{x}^*(t) to be linear in t per segment, facilitating sensitivity analysis and explicit solutions without re-solving the LP for each t. For example, in a one-parameter cost vector \mathbf{c}(t) = \mathbf{c}_0 + t \mathbf{c}_1, the optimal solution traces piecewise linear paths in the feasible region, changing at breakpoints where bases switch. Modern extensions to multi-parametric LP under global uncertainty partition the parameter space into polyhedral regions, each with a unique affine solution \mathbf{x}(t) = H_k t + G_k, enabling explicit controllers in model predictive control.^[107] More broadly, in nonlinear parametric optimization \min \{ f(\mathbf{x}, t) \mid g(\mathbf{x}, t) \leq \mathbf{0} \}, local optimal solutions \mathbf{x}(t) form differentiable curves under regularity conditions like the linear independence constraint qualification (LICQ) and second-order sufficient conditions, implicitly defined by the Karush-Kuhn-Tucker (KKT) system and solvable via the implicit function theorem as parametric equations \mathbf{x}(t). This parametrization supports stability analysis, such as Lipschitz continuity of \mathbf{x}(t) near nominal points, crucial for robust optimization in engineering and economics.^[107]

References

[1]
Parametric Equations -- from Wolfram MathWorld
Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as parameters.Missing: authoritative sources
[2]
7.1 Parametric Equations - Calculus Volume 2 | OpenStax
Mar 30, 2016 · The equations that are used to define the curve are called parametric equations. ... Many plane curves in mathematics are named after the ...
[3]
https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/11:_Parametric_Equations_and_Polar_Coordinates/11.01:_Parametric_Equations
[4]
Parametric Equations - Department of Mathematics at UTSA
Jan 15, 2022 · In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters.
[5]
Calculus II - Parametric Equations and Curves
Apr 10, 2025 · The collection of points that we get by letting t t be all possible values is the graph of the parametric equations and is called the parametric ...Missing: authoritative sources
[6]
https://www.math.washington.edu/~aloveles/Math126Winter2016/10-1%20and%2013-1%20Overheads%20(Landscape).pdf
[7]
[PDF] Parametric Curves - Vector Valued Functions - Trinity University
A single-variable vector function or parametric curve has the form r(t) = hx(t),y(t)i or r(t) = hx(t),y(t),z(t)i,
[8]
10.2 Parametric Equations‣ Chapter 10 Curves in the Plane ...
The graph of the parametric equations and is the set of all points ( x , y ) = ( f ⁢ ( t ) , g ⁢ ( t ) ) in the Cartesian plane, as the parameter t
[9]
Hippias - Biography - MacTutor - University of St Andrews
It was probably about 420 BC that Hippias of Elis invented the curve known as the quadratrix for the purpose of trisecting any angle.Missing: parametric | Show results with:parametric
[10]
[PDF] A History of Curves and Surfaces in CAGD - FarinHansford.com
Introduction. The term CAGD was coined by R. Barnhill and R. Riesenfeld in 1974 when they organized a conference on that topic at the University of Utah.
[11]
https://math.libretexts.org/Bookshelves/Algebra/Algebra_and_Trigonometry_1e_(OpenStax)/10:_Further_Applications_of_Trigonometry/10.06:_Parametric_Equations
[12]
[PDF] Parametric Curves - Texas Computer Science
Advantages of parametric forms. More degrees of freedom. Directly transformable. Dimension independent. No infinite slope problems.
[13]
[PDF] Parametric Curves & Surfaces - cs.Princeton
• A: Splines have several of advantages: • Numerically more stable. • Easier to compute. • Fewer bumps and wiggles. Page 4. 4. Catmull-Rom splines. • Properties.
[14]
[PDF] Conversion Methods between Parametric and Implicit Curves and ...
Oct 17, 1990 · For our purposes, projective curve parameterization has the advantage that all points on a parametric curve can be reached with finite parameter ...
[15]
CC Parametric Equations
Parametric equations are equations that specify the values of x and y in terms of a third variable t called a parameter.
[16]
10.4 Parametric Equations
Parametric equations are x=f(t) and y=g(t), where t is a parameter. They describe a curve, and can be used to describe an object's position over time.
[17]
[PDF] Parametric Curves
Here ˙α(t) = q and α(k) = 0 for k > 1. A polygon, on the other hand, is a piecewise smooth curve, where each edge determines a subdomain. x.
[18]
Calculus II - Arc Length with Parametric Equations
Aug 13, 2025 · In this section we will look at the arc length of the parametric curve given by, x=f(t)y=g(t)α≤t≤β We will also be assuming that the curve is traced out ...
[19]
[PDF] 7.2 | Calculus of Parametric Curves
To develop a formula for arc length, we start with an approximation by line segments as shown in the following graph. Chapter 7 | Parametric Equations and Polar ...
[20]
Curvature - Calculus III - Pauls Online Math Notes
Nov 16, 2022 · The curvature measures how fast a curve is changing direction at a given point. There are several formulas for determining the curvature for a curve.
[21]
[PDF] curvature.pdf
The (signed) curvature of a curve parametrized by its arc length is the rate of change of direction of the tangent vector. The absolute value of the curvature ...
[22]
[PDF] Math 162A - Introduction to Differential Geometry
A parametrized curve has an orientation (indicated by the blue arrow): as t increases along the interval. I, the point x(t) moves in a particular direction ...
[23]
[PDF] Computing parametrizations oF rational algebraic curves
Jun 1, 1994 · The theory of algebraic curves tells us that a curve can be parametrized precisely when the curve is rational, i.e. when the genus of the curve ...
[24]
[PDF] An Introduction to Parametrizing Rational Curves - UC Berkeley math
Dec 1, 2011 · We have showed how to obtain a rational parametrization for any given curve of genus 0. However, this algorithm may be quite slow for large ...
[25]
https://math.libretexts.org/Bookshelves/Calculus/Map%3A_Calculus__Early_Transcendentals_(Stewart)/13%3A_Vector_Functions/13.01%3A_Vector_Functions_and_Space_Curves
[26]
Serret-Frenet equations - PlanetMath
Mar 22, 2013 · relations, called the Serret-Frenet equations, hold between these three vectors. ... N′(s)⋅T(s)=−T′(s)⋅N(s)=−κ(s). ... N′(s)⋅N(s)=0. ... N′(s)⋅B(s)=τ( ...
[27]
Helix -- from Wolfram MathWorld
In fact, Lancret's theorem states that a necessary and sufficient condition for a curve to be a helix is that the ratio of curvature to torsion be constant.
[28]
CSC 418: Parametric Curves - Dynamic Graphics Project
Hermite curve segments can be connected in a continuous fashion by ensuring that the end-point of one curve is the same as the starting point of another, as ...
[29]
[PDF] differential-geometry-2024.pdf - Harvard Mathematics Department
It is convenient however to look first at manifolds embedded in a Euclidean space Rn like our space R3. Examples are one dimensional manifolds called curves ...
[30]
[PDF] Differentiable manifolds
Clearly, a parametrized manifold with m = 2 and n = 3 is the same as a parametrized surface, and the notion of regularity is identical to the one introduced in ...
[31]
Hypersphere -- from Wolfram MathWorld
The n-hypersphere (often simply called the n-sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers ...
[32]
[PDF] A parametrization algorithm to compute lower dimensional elliptic ...
Jan 29, 2025 · We present an algorithm for the construction of lower dimensional elliptic tori in parametric Hamiltonian systems by means of the ...
[33]
Analysis of parametric models | Advances in Computational ...
Nov 28, 2019 · From the factorisation, one obtains tensor representations, which may be cascaded, leading to tensors of higher degree. When carried over to a ...Missing: curves programming
[34]
[PDF] Making Mathematics Connections through Conic Sections Richard ...
Parametric Representations of the Conic Sections. The Pythagorean trigonometric identities allow for easy parametric representation of ellipses and hyperbolas.
[35]
[PDF] Tactile recognition of algebraic curves using semi-differential ...
Then, after some reparametrization and translation the equations of parabola reduces to its simplest form: x = at2, y = 2at, where a _ (—al sin B + bl cos 8)2 ...
[36]
https://dr.lib.iastate.edu/bitstreams/98bb65bb-8c6c-4f8d-a9ec-98a98b461053/download
[37]
10.0 Chapter Prerequisites — Conic Sections
A parabola is the locus of all points equidistant from a point (called a focus) and a line (called the directrix) that does not contain the focus.
[38]
[PDF] HYPERBOLA ELLIPSE PARABOLA MORE REVIEW OF CONIC ...
For both the ellipse and the hyperbola, e=c/a. The distance , p, from the focus to the conic section directrix of a conic section is called the focal parameter.
[39]
[PDF] Stereographic Projection in the Plane - College of the Holy Cross
stereographic projection (away from N). The inverse map π−1. N : R → S1 \ {N} is called the rational parameterization of the circle. ... called stereographic ...
[40]
[PDF] 1 Parametrized curve
The ellipse can also be given by a simple parametric form analogous to that of ... x = a cos t, y = b sin t, t ∈ (0, 2π). Note that cos2 t + sin2 t = 1. 5 ...
[41]
14. Mathematics for Orbits: Ellipses, Parabolas, Hyperbolas
The Hyperbola The semi-latus rectum, as for the earlier conics, is the perpendicular distance from a focus to the curve, and is ℓ=b2/a=a(e2−1). Each focus has ...
[42]
[PDF] Exploring Conics with Graphing Technology
All conic sections have an equation in general form given by: 0. 2. 2. = +. +. + ... = − AC. B then the figure is a parabola. Now let us turn our attention to ...<|control11|><|separator|>
[43]
Cycloid -- from Wolfram MathWorld
The evolute and involute of a cycloid are identical cycloids. If the cycloid has a cusp at the origin and its humps are oriented upward, its parametric equation ...
[44]
Epicycloid -- from Wolfram MathWorld
epicycloid is therefore an epitrochoid with h=b. Epicycloids are given by the parametric equations x = (a+b)cosphi-bcos((a+b)/bphi) (1) y = (a+b)sinphi-bsin ...
[45]
Hypocycloid -- from Wolfram MathWorld
x(0)=rho , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are. x, = (a-b)cosphi-bcostheta. (3). = (a-b) ...<|control11|><|separator|>
[46]
Hypotrochoid -- from Wolfram MathWorld
circle. The parametric equations for a hypotrochoid are x = (a-b)cost+hcos((a-b)/bt) (1) y = (a-b)sint-hsin((a-b)/bt), (2) A polar equation can be derived ...
[47]
Brachistochrone Problem -- from Wolfram MathWorld
In fact, the solution, which is a segment of a cycloid, was found by Leibniz, L'Hospital, Newton, and the two Bernoullis. Johann Bernoulli solved the problem ...
[48]
Lissajous Curve -- from Wolfram MathWorld
Lissajous curves are the family of curves described by the parametric equations x(t) = Acos(omega_xt-delta_x) (1) y(t) = Bcos(omega_yt-delta_y), ...
[49]
[PDF] arXiv:1802.06546v2 [math.NA] 20 Jul 2018
Jul 20, 2018 · The goal of this article is to provide a mathematical analysis of spherical Lissajous curves and to study their role as generating curves for ...
[50]
Prove Lissajous curves are closed given the ratio between a and b ...
Jun 8, 2020 · Viewed 759 times. 0. Lissajous curves are curves described by the parametric equations: x(t)=Asin(at+k), y(t)=Bsin(bt) According to wikipedia, ...
[51]
Parameterized Knots - Adelphi University
Abstract: We discuss two families of parameterized knots (polynomial and trigonometric) and include an interactive gallery of selected knots and their equations ...Missing: paper | Show results with:paper
[52]
Parametrizing some interesting knotted surfaces - arXiv
Jul 25, 2025 · Our main focus is to provide parametric equations, so in each case, we have attempted only those classes of knotted surfaces which arise ...
[53]
[PDF] Fourier Descriptors for Plane Closed Curves
A curve is represented parametrically as a function of arc length by the accumulated change in direction of the curve since the starting point.
[54]
Fourier-like expansion of a closed curve in 2D - Math Stack Exchange
Nov 18, 2011 · Fourier expansion can be used to represent any periodic function in one variable. Closed surfaces in 3D can be built out of spherical harmonics.mathbb R → \mathbb R^n$ is a parametrisation by arc-length of a ...Fourier series of almost periodic functions and regularityMore results from math.stackexchange.com
[55]
Viviani's Curve -- from Wolfram MathWorld
Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius a and center (a,0)
[56]
Calculus III - Parametric Surfaces - Pauls Online Math Notes
Mar 25, 2024 · In this section we will take a look at the basics of representing a surface with parametric equations.
[57]
[PDF] Parametric Surfaces and Surface Area - Purdue Math
Those three real valued functions are called parametric equations. Definition 2. A parametric surface is the image of a domain D in the uv plane under a.
[58]
[PDF] Differential Geometry of Surfaces - People @EECS
Differential geometry of a 2D manifold or surface embedded in 3D is the study of the intrinsic properties of the surface as well as the ef- fects of a given ...
[59]
3.2 First fundamental form I (metric) - MIT
The first fundamental form is defined as $\displaystyle I = ds^2 = d{\bf r} \cdot d{\bf r} = E du^2 + 2Fdudv + Gdv^2\;,$ and $ E$
[60]
[PDF] Basics of the Differential Geometry of Surfaces - CIS UPenn
For this, we will need to compute the norm of the tangent vector to a curve on a surface. This will lead us to the first fundamental form. 20.3 The First ...
[61]
[PDF] Gaussian and Mean Curvatures∗
Let κ1 and κ2 be the principal curvatures of a surface patch σ(u, v). The Gaussian curvature of σ is. K = κ1κ2, and its mean curvature is. H = 1.
[62]
Orienting surfaces - Math Insight
We can parametrize a surface in two different orientations. The orientation of a curve is given by the unit tangent vector n; the orientation of a surface is ...
[63]
[PDF] Orientations - Purdue Math
Definition 2. An orientation on an orientable surface is a choice of a continuous unit normal vector field.
[64]
[PDF] CS 468 [2ex] Differential Geometry for Computer Science
Theorem: Let S be a surface. For every p ∈ S there exists an isothermal parametrization for a neighbourhood of p. • This means that there exists U ⊆ R2 and ...
[65]
[PDF] Geometry of Surfaces - University of Utah Math Dept.
Jan 22, 2010 · Local Coordinates. A surface can locally be given by a curvilinear coordinate chart, also called a parameterization. Let U ⊂ R2 be open.
[66]
[PDF] Ruled surfaces and developable surfaces
Ruled surfaces are traced by a straight line. Developable ruled surfaces, or torses, have a single tangent plane along each ruling.<|separator|>
[67]
One-Sheeted Hyperboloid -- from Wolfram MathWorld
The one-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci.
[68]
Bridging the gap between rectifying developables and tangent ...
Feb 10, 2021 · The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the ...
[69]
Investigation of ruled surfaces and their singularities according to ...
In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space.Missing: challenges | Show results with:challenges
[70]
Chapter 31. Modeling and Processing with Quadric Surfaces
Quadric surfaces are surfaces defined by algebraic equations of degree two. It explains why quadrics are widely used in computer-aided geometric design (CAGD).
[71]
[PDF] Chapter 31 Modeling and rocessing with Quadric Surfaces
Quadric surfaces, or quadrics, are surfaces defined by algebraic equations of degree two. We will discuss mainly quadrics in 3D space, though quadrics in ...
[72]
http://i2.cs.hku.hk/~gfxgroup/publications/wenping-cagd.pdf
[73]
[PDF] Conversion Between Implicit and Parametric Representations of ...
In this chapter, we review algorithms for conversion between implicit and paramet- ric representations of algebraic varieties with emphasis on the ...
[74]
Implicitization of rational parametric equations - ScienceDirect.com
Based on the Gröbner basis method, we present algorithms for a complete solution to the following problems in the implicitization of a set of rational ...
[75]
[PDF] Implicitizing Rational Parametric Surfaces - UC Berkeley EECS
In Section 3 we analyze the problem of implicitization and show how we can use resultants or Grőbner bases on a parametrization without any base points to ...
[76]
https://mathworld.wolfram.com/SphericalCoordinates.html
[77]
Parametric equations - Ximera - The Ohio State University
Use this method to find a parametric equation for a line segment that starts ... 6 Converting from parametric to cartesian. We mentioned earlier that ...
[78]
3.3 Arc Length and Curvature
Feb 3, 2025 · Any smooth curve can be expressed with arc length as parameter: , d s / d t = ‖ r → ′ ( t ) ‖ > 0 , so s ( t ) is increasing and has an inverse, ...
[79]
[PDF] 18.01 Calculus Jason Starr Fall 2005
Nov 3, 2005 · This gives the formula for the arc length of an explicit curve, b Arc length = 1 + (dy/dx)2dx.
[80]
[PDF] Parametric Curves, Polar Coordinates, and Complex Numbers
Definition: If x(t) and y(t) are functions of t, then the set of points (x, y)=(x(t), y(t)) is called a parametric curve. The points on this curve are traced ...
[81]
[PDF] Parameterization and Applications of Catmull-Rom Curves
Aug 23, 2010 · Finally, Catmull-Rom curves have an explicit piecewise polynomial representation, allowing them to be easily be converted to other bases and.
[82]
[PDF] Spline Toolbox User's Guide
efficient means for converting from ppform to B-form. Constructive vs ... Explicitly, such a form describes a function in terms of the basis function ...
[83]
[PDF] Chapter 8 Rational Parametrization of Curves
In the next sections we will show how to check the rationality by algorithmic methods and how to actually compute rational parametrizations of algebraic curves.
[84]
[PDF] pythagorean triples - keith conrad
A Pythagorean triple is a triple of positive integers (a, b, c) where a² + b² = c². Examples include (3,4,5), (5,12,13), and (8,15,17).
[85]
[PDF] Introduction to Pythagorean-hodograph curves
It is impossible to parameterize any plane curve, other than a straight line, by rational functions of its arc length. rational parameterization r(t)=(x(t),y(t)) ...
[86]
circular helix - PlanetMath.org
Mar 22, 2013 · A circular helix can be conceived of as a space curve with constant, non-zero curvature, and constant, non-zero torsion.
[87]
[PDF] Motion Along a Curve
1 Sketch the curve with parametric equations x = t, y = t3. Find the velocity vector and the speed at t = 1. 2 Sketch the path with parametric equations x = 1 ...
[88]
10.7: Parametric Equations- Graphs - Mathematics LibreTexts
Apr 27, 2025 · Projectile motion depends on two parametric equations: $x=(v_0 \cos \theta)t$ and $y=−16t^2+(v_0 \sin \theta)t+h$. Initial velocity is ...Learning Objectives · Graphing Parametric... · Exercise $\PageIndex{3}$
[89]
4.3 Projectile Motion - University Physics Volume 1 | OpenStax
Sep 19, 2016 · h = v 0 y 2 2 g . This equation defines the maximum height of a projectile above its launch position and it depends only on the vertical ...
[90]
3.3: Projectile Motion - Physics LibreTexts
Nov 5, 2020 · The range of an object, given the initial launch angle and initial velocity is found with: R = v i 2 ⁢ sin ⁡ ⁡ 2 ⁢ θ i g . The maximum height of ...Projectile Motion · Solving Problems · Zero Launch Angle · General Launch Angle
[91]
Harmonic motion
An object moving along the x-axis is said to exhibit simple harmonic motion if its position as a function of time varies as. x(t) = x0 + A cos(ωt + φ).
[92]
15.3: Energy in Simple Harmonic Motion - Physics LibreTexts
Sep 12, 2022 · The energy in a simple harmonic oscillator is proportional to the square of the amplitude. When considering many forms of oscillations, you will ...Missing: parametric | Show results with:parametric
[93]
[PDF] A geometrical introduction to screw theory - arXiv
Nov 20, 2012 · This work introduces screw theory, a venerable but yet little known theory aimed at describing rigid body dynamics.
[94]
[PDF] Screw Theory, Exponential Coordinates, and Forward Kinematics
Feb 7, 2023 · Chasles-Mozzi Theorem: Any displacement in 3D can be represented by a rotation and translation about the same axis, referred to as a screw ...
[95]
[PDF] World Lines - UCSB Physics
A world-line is the curve in space-time that indicates a particle's trajectory. Its space-time length equals the proper time experienced.
[96]
[PDF] A SPACETIME PRIMER
Sep 2, 2004 · The proper time between two events is maximized by the inertial (i.e., straight in inertial coordinates) world line that connects them. Since an ...
[97]
[PDF] Bézier Curves (1968-1977)
Bézier's PhD Thesis (in maths) on the study of parametric polynomial curves and their vector coefficients. Bézier presented his PhD thesis in 1977, after his.Missing: seminal paper
[98]
[PDF] On NURBS: a survey - IEEE Computer Graphics and Applications
NURBS are genuine generalizations of nonrational B- spline forms as well as rational and nonrational Bezier curves and surfaces. However, NURBS have several ...Missing: seminal | Show results with:seminal
[99]
[PDF] Ray Tracing Parametric Surfaces
In this paper we present a new algorithm for finding the intersection of a ray and parametric surface. In order to provide for complete gen- erality in the ...<|separator|>
[100]
Chapter 42. Deformers - NVIDIA Developer
A deformer is an operation that takes as input a collection of vertices and generates new coordinates for those vertices (the deformed coordinates).Missing: parametrizations | Show results with:parametrizations
[101]
Parametric Form
The parameteric form is much more explicit: it gives a concrete recipe for producing all solutions.
[102]
https://fab.cba.mit.edu/classes/S62.12/docs/Piegl_NURBS.pdf
[103]
[PDF] Lectures on Parametric Optimization: An Introduction
The present report aims to provide an overview of results from parametric opti- mization which could be called classical results on the subject.