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Homothety

In mathematics, a homothety (also known as a dilation) is a geometric transformation of an affine space, defined by a fixed point called the center and a nonzero real number k called the scaling factor or ratio, which maps every point P to a point P' such that the vector from the center O to P' is k times the vector from O to P, or \overrightarrow{OP'} = k \cdot \overrightarrow{OP}.^[1] This transformation enlarges the figure if |k| > 1, reduces it if $0 < |k| < 1, or inverts it relative to the center if k < 0.^[2] Homotheties are fundamental in Euclidean geometry for establishing similarities between figures, as they preserve angles, parallelism of lines, and the ratios of distances not involving the center, while scaling all distances from the center by |k|.^[3] Key properties of homotheties include their composition: the composition of two homotheties with the same center and ratios k_1 and k_2 yields another homothety with ratio k_1 k_2, and they are invertible via a homothety with ratio $1/k.^[4] For k > 0, homotheties are orientation-preserving, maintaining the handedness of figures, whereas for k < 0, they reverse orientation, effectively combining scaling with a point reflection.^[5] The center O is the only fixed point unless k = 1, in which case the transformation is the identity.^[2] Homotheties play a crucial role in solving geometric problems, such as proving concurrency of lines, collinearity of points, and properties of similar triangles, by mapping one configuration to another while preserving essential relations.^[6] They also appear in broader contexts, including the study of conic sections, where homothetic transformations relate different conics sharing a common center, and in affine geometry as a building block for similarities, which combine homotheties with isometries.^[4] In higher dimensions, homotheties extend naturally to vector spaces, facilitating analysis in linear algebra and computer graphics for scaling operations.^[3]

Definition and Fundamentals

Formal Definition

A homothety, also known as a dilation or central similarity, is a geometric transformation in Euclidean space that maps every point P to a point P' lying on the line through a fixed center O and P, such that the directed distance satisfies OP' = k \cdot OP, where k \in \mathbb{R} \setminus \{0\} is the scaling factor.^[7] If k > 0, P' lies on the ray starting at O and passing through P (direct homothety, preserving orientation); if k < 0, P' lies on the opposite ray from O in the direction away from P (opposite homothety, reversing orientation).^[7] In vector notation, the homothety H with center O and scaling factor k is expressed as

H(P) = O + k (P - O).

^[7] This formulation highlights that O is the unique fixed point, as H(O) = O.^[4] A homothety is a special case of a similarity transformation, distinguished by its fixed center O and uniform scaling applied radially from O, whereas general similarities compose homotheties with isometries like rotations or translations.^[4] For instance, with k = 2, distances from O are doubled, expanding figures outward; with k = 1/2, distances are halved, contracting figures toward O.^[7]

Center and Scaling Factor

In a homothety, the center O serves as the unique fixed point, satisfying H(O) = O, which remains invariant under the transformation. This point acts as the origin for the scaling operation, ensuring that all rays emanating from O are preserved in their direction, though their lengths are altered proportionally. The center O is pivotal in defining the transformation's geometry, as it determines the focal point around which the dilation occurs.^[7]^[8] The scaling factor k, a real number, governs the magnification or contraction effected by the homothety. When |k| > 1, the transformation expands figures away from the center; conversely, $0 < |k| < 1 contracts them toward it. If k > 0, orientation is preserved, whereas k < 0 reverses it, combining dilation with reflection. Special cases include k = 1, which yields the identity transformation. For k \neq 1, the center O is the sole invariant point. The inverse of a homothety with factor k is another homothety sharing the same center but with scaling factor $1/k.^[7]^[8]^[9] Geometrically, a homothety can be interpreted as a radial stretching or shrinking from the center O by the factor k, formally expressed as H(P) = O + k (P - O) for any point P. This radial action maintains the directional integrity of lines through O while scaling distances from it uniformly.^[7]^[8]^[9]

Geometric Properties

Mapping of Lines, Segments, and Angles

A homothety with center O and scaling factor k \neq 0 maps lines in the Euclidean plane to lines, with specific behavior depending on their relation to the center. Any line passing through O is mapped to itself, as every point on the line lies on a ray from O and is scaled along that ray, preserving the line's position and direction.^[6] For lines not passing through O, the image is a line parallel to the original, since the transformation scales vectors from O uniformly, maintaining the direction of the line's displacement vector while shifting its position.^[10] This parallelism holds regardless of the value of k, as the homothety acts as an affine transformation that preserves directional relationships away from the center.^[9] Line segments are transformed similarly under homothety. A segment AB not containing O maps to a segment A'B' that is parallel to AB, with its length scaled by the absolute value |k|, such that |A'B'| = |k| \cdot |AB|.^[6] The endpoints A and B are mapped along the rays from O through them, ensuring the vector \overrightarrow{A'B'} = k \cdot \overrightarrow{AB}, which confirms both the parallelism and the scaling.^[10] If the segment passes through O, its image is another segment on the same line, with length scaled by |k|. The scaling factor k determines the uniform ratio for lengths measured from points away from O. Angles are preserved in measure by a homothety, reflecting its role as a similarity transformation. For an angle \angle XYZ with vertex at Y, the image \angle X'Y'Z' has the same measure as \angle XYZ, since the rays YX and YZ map to rays from Y' that maintain the original angular separation due to uniform scaling of directions from O.^[10] However, if k < 0, the homothety reverses orientation, transforming a counterclockwise angle to clockwise (or vice versa), while still preserving the absolute measure.^[6] The vertex Y maps to Y' along the ray from O through Y. Homotheties map circles to circles and lines to lines (either the same line if passing through O or a parallel line otherwise), which relates to their preservation of conic sections.^[11]

Preservation of Incidence and Collinearity

A homothety, being an affine transformation, preserves the incidence structure of points and lines in the plane. Specifically, if a point lies on a given line, then its image under the homothety lies on the image of that line, ensuring that the transformation maps the incidence relation bijectively.^[12] This property follows from the fact that homotheties are collineations, which by definition map lines to lines while maintaining point-line incidences.^[13] Homotheties also preserve collinearity, meaning that any set of collinear points maps to another set of collinear points. As a type of similarity transformation, a homothety ensures that if points A, B, and C are collinear, then their images H(A), H(B), and H(C) remain collinear, with the order and relative positions adjusted according to the scaling factor k.^[14] Furthermore, homotheties preserve parallelism: parallel lines are mapped to parallel lines, regardless of whether they pass through the center of homothety, due to their affine nature.^[12] For lines passing through the center O, the image coincides with the original line, while parallel lines not through O are scaled and translated but retain their parallel orientation.^[13] Along any line, a homothety maintains directed ratios in which points divide segments, though the actual distances are scaled by the factor k. For collinear points A and B, and a point P dividing the segment AB in the ratio \lambda : 1 - \lambda (in affine sense), the image H(P) divides H(A)H(B) in the same ratio \lambda : 1 - \lambda, preserving the barycentric coordinates relative to the line.^[12] The directed distances between images are multiplied by k, so ratios of directed segments, such as \frac{H(P)H(A)}{H(P)H(B)}, adjust accordingly to reflect the scaling while keeping the proportional division invariant.^[15] This ratio preservation is key to applications in proving geometric theorems involving proportional segments.

Constructions and Visualizations

Intercept Theorem Application

The intercept theorem, also known as Thales' theorem or the basic proportionality theorem, states that if a line parallel to one side of a triangle intersects the other two sides, then it divides those sides in the same ratio.^[16] This theorem, formalized in Euclid's Elements as Book VI, Proposition 2, provides the geometric foundation for constructing proportional segments, which is crucial for determining the positions of image points in a homothety.^[16] By applying parallels and transversals, the theorem enables the scaling of distances from the homothety center O without direct measurement, ensuring the image figure maintains similarity to the original. To construct the image of a figure ABC under a homothety with center O and scaling factor k = 2, begin by drawing the rays from O through each vertex A, B, and C. For the image point A' on ray OA such that OA' = 2 · OA, apply the intercept theorem as follows: Draw an auxiliary ray from O in a direction not coinciding with OA. Mark two equal segments on this auxiliary ray, locating points S (after the first segment) and T (after the second segment), so OS = ST. Connect S to A, forming transversal SA. Then, through T, draw a line parallel to SA, intersecting ray OA at A'. By the intercept theorem, the parallel line divides the transversals proportionally, yielding OA' / OA = OT / OS = 2 / 1.^[17] Repeat this process for points B' on ray OB and C' on ray OC. The triangle A'B'C' is the homothetic image of ABC, similar with ratio 2. This method leverages the preservation of parallelism under homothety, as corresponding lines in the original and image figures remain parallel.^[10] As an example, consider constructing the image of a square ABCD with center O and k = 2. Draw rays from O through vertices A, B, C, and D. Using the intercept theorem on each ray as described, locate the image points A', B', C', and D'. Connecting these points yields a square A'B'C'D' enlarged by factor 2, with sides parallel to the original and distances from O doubled.^[17]

Pantograph Mechanism

The pantograph is a mechanical linkage device constructed as a four-bar mechanism utilizing parallelogram arms, enabling the tracer point to replicate the motion of a drawing point on a scaled basis.^[18]^[19] In this setup, four rigid rods are connected by pivoting joints to form a parallelogram configuration, with extensions allowing one end to guide a stylus while the opposite end holds a pen that traces an enlarged or reduced copy of the original path.^[20] The device embodies a homothety with the fixed pivot serving as the center of scaling, denoted as point O, where the ratio of the arm lengths dictates the scaling factor k, facilitating consistent enlargement or reduction of figures by a predetermined multiple.^[20]^[21] This mechanical realization ensures that distances from the pivot are multiplied by k, mirroring the geometric transformation of homothety while maintaining shape integrity. Invented by German Jesuit astronomer Christoph Scheiner between 1603 and 1605, the pantograph was initially developed to copy and scale astronomical diagrams accurately.^[22]^[23] Over time, it found extensive application in cartography for enlarging or reducing maps and in drafting for precise technical reproductions.^[24] In operation, motion applied at the input end—such as guiding the tracer along a curve—propagates through the linkage geometry to produce an output path at the drawing end, scaled uniformly by the arm ratio while preserving angles due to the parallelogram's parallel motion.^[20] This angle preservation aligns with the properties of homothety, ensuring that the replicated figure remains similar to the original without distortion.^[18]

Algebraic Formulations

Cartesian Coordinate Representation

In the Euclidean plane equipped with Cartesian coordinates, a homothety with center O = (a, b) and scaling factor k \neq 0 maps a point P = (x, y) to its image H(P) = (x', y'), where the coordinates satisfy the vector equation \mathbf{H(P)} = \mathbf{O} + k (\mathbf{P} - \mathbf{O}).^[10] Expanding this component-wise yields the explicit formulas:

x' = a + k (x - a), \quad y' = b + k (y - b).

This representation follows directly from the geometric definition of homothety, which scales vectors from the center by the factor k, preserving directions and collinearity while altering distances proportionally.^[25] Equivalently, the transformation can be expressed in affine form as \mathbf{H(P)} = k \mathbf{P} + (1 - k) \mathbf{O}, highlighting its structure as a linear scaling combined with a translation.^[25] In matrix notation, this corresponds to the affine transformation

\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} + (1 - k) \begin{pmatrix} a \\ b \end{pmatrix},

where the diagonal matrix represents uniform scaling by k about the origin, adjusted by the translation vector to shift the fixed point to O.^[10] This matrix form underscores homothety's role as a special case of affine transformations, with the scaling factor k determining expansion (k > 1) or contraction ($0 < k < 1).^[25] For illustration, consider a homothety centered at the origin O = (0, 0) with k = 3 applied to P = (1, 2). The image is H(P) = (3 \cdot 1, 3 \cdot 2) = (3, 6). The distance from the center scales accordingly: the original distance \sqrt{1^2 + 2^2} = \sqrt{5} maps to $3\sqrt{5}, confirming the factor k = 3.^[10]

Homogeneous Coordinate Representation

In projective geometry, homotheties are represented using homogeneous coordinates, where a point in the affine plane is denoted as [x : y : 1] and the center O as [a : b : 1]. This setup allows the homothety to be formulated as a linear transformation on the projective plane \mathbb{RP}^2, extending the affine case by incorporating the line at infinity.^[26] The matrix representation of a homothety with center (a, b) and scaling factor k \neq 0 in homogeneous coordinates is given by

\begin{pmatrix} k & 0 & a(1 - k) \\ 0 & k & b(1 - k) \\ 0 & 0 & 1 \end{pmatrix}.

This matrix is derived by composing a translation to move the center to the origin, a uniform scaling by k via the diagonal matrix \operatorname{diag}(k, k, 1), and a translation back to the original center, all unified in the projective framework.^[27]^[28] For the special case where the center is at the origin O = [0 : 0 : 1], the matrix simplifies to \operatorname{diag}(k, k, 1), which directly scales a point [x : y : 1] to [k x : k y : 1]. This homogeneous representation offers key advantages in projective geometry: it naturally handles points at infinity, as the third coordinate can be zero without singularity, allowing homotheties to map parallel lines (intersecting at infinity) to parallel lines while preserving the projective structure.^[26] Additionally, when k < 0, the transformation inherently includes orientation reversal in the projective plane, treating negative scalings as antihomotheties without special affine adjustments.^[10] This contrasts with the purely affine Cartesian coordinate representation, which is a special case limited to finite points.^[27]

Composition and Extensions

Combining Multiple Homotheties

The composition of two homotheties H_1 with center O_1 and ratio k_1, followed by H_2 with center O_2 and ratio k_2, results in a transformation with ratio k_1 k_2. If k_1 k_2 \neq 1, this composition is another homothety whose center lies on the line joining O_1 and O_2. If k_1 k_2 = 1, the composition is a translation in the direction parallel to the vector from O_1 to O_2.^[10] When O_1 = O_2 = O, the composition is a homothety with the same center O and ratio k_1 k_2. In this case, the homotheties commute under composition.^[10] For distinct centers, the center O of the composite homothety divides the segment O_1 O_2 in a specific ratio. Considering the order H_2 \circ H_1, let the first homothety have center A = O_1 and ratio r = k_1, and the second have center B = O_2 and ratio s = k_2; then O divides AB such that the ratio AO : OB = (1 - s) : [s (1 - r)], provided r s \neq 1. Equivalently, in vector notation, the position of O is given by the barycentric combination

\mathbf{O} = \frac{s(1 - r) \mathbf{A} + (1 - s) \mathbf{B}}{s(1 - r) + (1 - s)}.

^[29]^[30] Iterating homotheties with the same center yields straightforward results: the composition of n such homotheties with ratios k_1, \dots, k_n is a single homothety with the common center and ratio \prod_{i=1}^n k_i. In particular, the n-th iterate H^n of a homothety H with ratio k has ratio k^n and the same center.^[10] The set of all homotheties sharing a fixed center forms an abelian group under composition, isomorphic to the multiplicative group of nonzero real numbers; the identity element is the homothety with ratio 1, and every element with ratio k \neq 0 has an inverse given by the homothety with ratio $1/k and the same center.^[10]

Relation to Similarity Transformations

A homothety is a special case of a similarity transformation, characterized by its fixed center of scaling, which distinguishes it from general similarities that combine uniform scaling with arbitrary rotations, translations, or reflections.^[4] While general similarity transformations preserve angles and ratios of distances but allow for orientation changes and displacements, a homothety maintains all points aligned radially from the fixed center, resulting in a pure dilation or contraction without additional rigid motions.^[31] The scale factor k of a homothety directly corresponds to the similarity ratio |k|, ensuring that distances from the center are multiplied by |k| while preserving shape and orientation.^[4] Every homothety qualifies as a similarity transformation with ratio |k|, and conversely, any similarity can be decomposed as the composition of an isometry (such as a rotation or translation) followed by a homothety of the same ratio.^[4] This composition property highlights how homotheties serve as the scaling component within the broader group of similarities, enabling the generation of all orientation-preserving similarities through combinations with direct isometries.^[32] For instance, precomposing a homothety with a rotation around its center yields a spiral similarity, a key subclass of similarities used in geometric problem-solving.^[31] In inversive geometry, homotheties play a role in preserving circles and spheres, as they map circles to concentric or parallel circles, aligning with the circle-preserving nature of inversions and their compositions.^[33] This property extends to fractal constructions, where repeated applications of homotheties with ratio $1/3 and rotations generate self-similar structures like the Koch snowflake, iteratively building the boundary from an initial equilateral triangle.^[34] In computer graphics, homotheties implement dilations for pivot-based scaling, allowing distortion-free enlargement or reduction of images relative to a fixed point without requiring full similarity computations.^[35]

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