Dilation Calculator
Calculate geometric dilation transformations with scale factors
Table of Contents
How to Use
- Enter the X coordinate of the center of dilation
- Enter the Y coordinate of the center of dilation
- Enter the X coordinate of the original point
- Enter the Y coordinate of the original point
- Enter the scale factor (k > 1 enlarges, 0 < k < 1 reduces, k < 0 reflects)
- Click Calculate to see the dilated point and transformation details
What is Dilation?
Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure based on a scale factor.
Every dilation has a center point and a scale factor k. The center is a fixed point in the plane about which all points are expanded or contracted. The scale factor describes how much the figure is enlarged or reduced.
Understanding Scale Factor
| Scale Factor (k) | Effect | Example |
|---|---|---|
| k > 1 | Enlargement | k = 2 doubles the distance from center |
| k = 1 | No change | Image is identical to original |
| 0 < k < 1 | Reduction | k = 0.5 halves the distance from center |
| k < 0 | Reflection and scaling | k = -1 reflects through center |
| k = 0 | Invalid | All points collapse to center (not allowed) |
Dilation Formula
The formula for dilation is: P' = C + k(P - C)
Where:
- P' = (x', y') is the dilated point
- C = (cx, cy) is the center of dilation
- k is the scale factor
- P = (x, y) is the original point
Expanded form: x' = cx + k(x - cx) and y' = cy + k(y - cy)
Properties of Dilation
- Preserves shape (similar figures)
- Preserves angle measures
- Does not preserve distance (unless k = 1 or k = -1)
- Parallel lines remain parallel
- The center of dilation is the only fixed point (unless k = 1)
- Distance from center is multiplied by |k|
- Area is multiplied by k²
- Volume (in 3D) is multiplied by k³
Real-World Applications
- Computer graphics and image scaling
- Architecture and scale models
- Map reading and scale drawings
- Photography and zoom functions
- Medical imaging (CT scans, X-rays)
- Engineering design and CAD software
- Animation and special effects
Frequently Asked Questions
- What happens when the scale factor is negative?
- A negative scale factor causes both a reflection through the center of dilation and a size change. For example, k = -2 reflects the point through the center and doubles its distance from the center.
- Can the center of dilation be at the origin?
- Yes, when the center is at the origin (0, 0), the dilation formula simplifies to P' = kP, making calculations easier. This is a common special case.
- Is dilation the same as scaling?
- Yes, dilation and scaling are the same transformation. Both terms describe enlarging or reducing a figure by a scale factor relative to a fixed point.
- What's the difference between dilation and translation?
- Dilation changes the size of a figure (and possibly reflects it), while translation moves a figure without changing its size or orientation. Dilation has a center point; translation has a direction and distance.