Center of Dilation
Geometry & MeasurementThe center of dilation is the fixed point from which all points of a figure are scaled outward or inward during a dilation.
Definition
The center of dilation is the fixed point that does not move when a dilation is performed. All other points move toward or away from the center. Lines connecting original points to their images all pass through the center.
Example
If you dilate a triangle from the origin by scale factor $3$, every vertex moves to a point $3$ times farther from the origin. Draw a line from the origin to vertex $A$ and continue it: the image $A'$ is on that line, $3$ times as far out.
Key Insight
The center of dilation is like the anchor point for resizing. Zoom in on a photo using a pinch gesture: the point you pinch from is the center of dilation. Everything expands or contracts from that fixed point.
Definition
The center of dilation $C$ is the unique fixed point of the dilation: $D(C,k)(C) = C$ for any scale factor $k$. For any point $P$, the image $P'$ lies on ray $CP$ (or its opposite ray if $k < 0$), at distance $|k| \cdot |CP|$ from $C$. To find the center from a figure and its image, draw lines through corresponding points; they all intersect at $C$.
Example
To find the center of dilation between triangle $ABC$ and its image $A'B'C'$: draw line $AA'$ and line $BB'$. The center is where they intersect. Then check that $CC'$ passes through the same point. If all three lines meet at one point, it is the center of dilation.
Key Insight
The center of dilation can be inside, outside, or on the original figure. If the center is at a vertex of the figure, that vertex maps to itself. The center is the "vanishing point" in perspective drawing, where parallel lines appear to meet.
Definition
The center of dilation is the unique fixed point of the dilation map $f(x) = C + k(x - C) = kx + (1-k)C$. Solving $f(x) = x$ gives $x = C$. In the complex plane, the center is the fixed point of the Mobius-like map $f(z) = kz + (1-k)C$, found by solving $f(z) = z$. For a composition of two dilations with different centers and $k_1 k_2 \neq 1$, the new center is found by intersecting the lines through corresponding image pairs.
Example
Composition of $D(C_1=0, k_1=2)$ and $D(C_2=(6,0), k_2=1/3)$: the composed scale factor is $2(1/3) = 2/3$. New center: solve $x = C_2 + (1/3)(C_1 + 2(x - C_1) - C_2)$, giving $x = (9,0)/2 = (4.5, 0)$.
Key Insight
In projective geometry, the center of dilation corresponds to the "center of perspectivity" in Desargues' theorem: two triangles are in perspective from a point if lines through corresponding vertices are concurrent. Dilation is the affine special case of this projective concept.