1. What is Dilation in Geometry?

Dilation is a transformation that produces an image that is the same shape as the original, but is a different size. It involves scaling all distances from a fixed point called the center of dilation.

2. How Does the Dilation Calculator Work?

The calculator uses the dilation formulas:

Where:

• \( x, y \) — Original coordinates of the point
• \( c_x, c_y \) — Coordinates of the center of dilation
• \( k \) — Scale factor (k > 1 enlarges, 0 < k < 1 reduces, k < 0 reflects and scales)
• \( x', y' \) — Coordinates of the dilated point

Explanation: The formula calculates the new position by moving the point away from or toward the center by a factor of k.

3. Understanding the Center of Dilation

Details: The center of dilation is the fixed point in the plane about which all points are expanded or contracted. When the center is at the origin (0,0), the formulas simplify to simple scaling.

4. Using the Calculator

Tips: Enter the original point coordinates, center of dilation coordinates, and scale factor. Positive scale factors maintain orientation, negative factors create a reflection.

5. Frequently Asked Questions (FAQ)

Q1: What does a scale factor of 1 mean?
A: A scale factor of 1 means no change - the dilated image is identical to the original.

Q2: How does negative scaling work?
A: Negative scaling creates a reflection across the center point while also scaling the distance.

Q3: What's the difference between dilation and translation?
A: Dilation changes size (and possibly reflects), while translation moves the object without changing its size or shape.

Q4: Can the center of dilation be outside the shape?
A: Yes, the center can be anywhere - inside, on the edge, or outside the shape being dilated.

Q5: How is this used in real-world applications?
A: Dilation is used in computer graphics, map scaling, architectural designs, and medical imaging.