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The Geometry Dilation Calculator handles dilation from any center point (not just the origin) by any nonzero scale factor. Where the generic Geometric Transformation Calculator assumes dilation centered at the origin (the easy case), this tool implements the full formula for dilation about an arbitrary point. This tutorial walks through what dilation does geometrically, derives the two-coordinate transformation rule, and shows how dilation relates to the broader concept of similarity.

What dilation does

Dilation is a transformation that scales every point of a figure away from (or toward) a fixed center by the same factor. After dilation:

• Angles are preserved — the figure keeps its shape.
• Lengths are multiplied by the scale factor k.
• Areas are multiplied by k².
• The center of dilation is the only fixed point — it does not move.

Dilation is the source of the entire concept of similarity: two figures are similar if one is a dilation of the other (possibly combined with a rotation or reflection).

The transformation rule

For dilation centered at C = (cx, cy) with scale factor k, a point P = (x, y) maps to P' = (x', y') where:

x' = cx + k(x − cx)
y' = cy + k(y − cy)

Where this comes from: the vector from C to P is (x − cx, y − cy). Scaling that vector by k gives (k(x − cx), k(y − cy)). Adding it back to C gives the image P'. In compact vector form: P' = C + k(P − C).

Special case — dilation from the origin

When C = (0, 0), the formula collapses to:

x' = 0 + k(x − 0) = kx
y' = 0 + k(y − 0) = ky

So (x, y) → (kx, ky). This is the simpler version found in most introductory geometry textbooks. Real-world problems often involve an arbitrary center, though, which is why this calculator implements the full rule.

What the scale factor k controls

Worked example 1 — Dilation from the origin

Point P = (4, 6), center C = (0, 0), scale factor k = 2.

x' = 0 + 2(4 − 0) = 8
y' = 0 + 2(6 − 0) = 12
P' = (8, 12)

The image is twice as far from the origin as the original, in the same direction.

Worked example 2 — Dilation from an arbitrary center

Point P = (4, 6), center C = (1, 2), scale factor k = 2.

x' = 1 + 2(4 − 1) = 1 + 6 = 7
y' = 1 + 2(6 − 2) = 2 + 8 = 10
P' = (7, 10)

Note that P' is NOT just (8, 12) — the center shifts the result. The image P' satisfies: vector from C to P' is exactly 2× the vector from C to P. Check: P − C = (3, 4), P' − C = (6, 8) — yes, doubled.

Worked example 3 — Reduction with center

Point P = (10, 10), center C = (4, 4), scale factor k = 0.5.

x' = 4 + 0.5(10 − 4) = 4 + 3 = 7
y' = 4 + 0.5(10 − 4) = 7
P' = (7, 7)

The image lies halfway between P and the center C — that's what scale factor 0.5 does.

Dilation of a whole figure

To dilate a polygon or curve, apply the same dilation rule to every point individually. For a polygon, you dilate each vertex and connect them in the same order. The result has:

• Same number of vertices
• Same angles at corresponding vertices
• All sides scaled by factor |k| (absolute value because negative k flips orientation but does not negate lengths)
• Area scaled by k² (always positive — area can't be negative regardless of k's sign)

How dilation produces similarity

Two figures are similar if one can be transformed into the other by some combination of dilation, rotation, reflection, and translation. The "k" in dilation IS the scale factor of similarity. If a triangle is dilated by k = 3, the image triangle is similar to the original with linear ratio 3 and area ratio 9.

This is why dilation is the only one of the four basic transformations (translation, reflection, rotation, dilation) that does NOT produce a congruent image — it produces a similar one. The three isometries all give congruence; dilation alone breaks the size constraint.

Real-world applications

• Map scaling. A map with scale 1:24,000 is a dilation of the real terrain by k = 1/24,000.
• Architectural drawings. A blueprint at 1/4" = 1' scale is a dilation by k = 1/48.
• Computer graphics zoom. Pinch-zoom on a phone is a dilation centered at the midpoint of your two fingers, with k = (current pinch distance) / (initial pinch distance).
• Microscope and telescope optics. Magnification is the absolute value of the dilation factor produced by the optical system, with the optical axis as the center.

Common mistakes

• Confusing dilation with translation. Translation slides every point by the same vector. Dilation scales every point relative to a fixed center — points further from the center move more.
• Forgetting to subtract the center before scaling. The formula is k × (point − center), not k × point. Forgetting the subtraction gives the wrong result whenever the center is not the origin.
• Negative scale factor confusion. Negative k means the image lies on the opposite side of the center from the original. It is NOT the same as a reflection across an axis.
• Assuming area scales linearly. Area scales as k², not k. Doubling lengths quadruples area. This is the same lesson as in similar polygons.