1. What is Figure Dilation?

Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. It stretches or shrinks the original figure by a scale factor relative to a fixed point called the center of dilation.

2. How Does the Dilation Formula Work?

The calculator uses the dilation formula:

Where:

• \( \text{new\_points} \) — Coordinates after dilation
• \( \text{old\_points} \) — Original coordinates
• \( \text{center} \) — Center of dilation coordinates
• \( k \) — Scale factor (k > 1 enlarges, 0 < k < 1 reduces)

Explanation: The formula calculates each new coordinate by finding the vector from the center to the original point, scaling it by k, and then adding it back to the center coordinates.

3. Importance of Dilation in Geometry

Details: Dilation is fundamental in similarity transformations, map scaling, computer graphics, and understanding proportional relationships in geometric figures.

4. Using the Calculator

Tips: Enter coordinates as x,y pairs (e.g., "3,5"). The scale factor k determines the size change: values greater than 1 enlarge the figure, between 0 and 1 shrink it, and negative values create an inversion.

5. Frequently Asked Questions (FAQ)

Q1: What happens when k = 1?
A: The figure remains unchanged as the dilation scale factor is 1.

Q2: How does negative k affect the dilation?
A: Negative k creates an inversion (like a mirror image) along with the size change.

Q3: What's the difference between dilation and scaling?
A: Dilation specifically refers to scaling relative to a fixed center point, while general scaling might not preserve the position relative to a center.

Q4: Can I dilate 3D figures with this formula?
A: The same principle applies in 3D - just extend the coordinates to include z-values.

Q5: How is dilation used in real-world applications?
A: Applications include computer graphics, map scaling, architectural blueprints, and medical imaging.