1. What is Point Dilation?

Dilation is a transformation that changes the size of a geometric figure while maintaining its shape. For a point, dilation moves it closer to or farther from a center point based on a scale factor.

2. How Does the Calculator Work?

The calculator uses the dilation formula:

Where:

• \( x' \) — Dilated point coordinate
• \( c_x \) — Center point coordinate
• \( k \) — Scale factor (unitless)
• \( x \) — Original point coordinate

Explanation: The formula calculates the new position of a point after scaling its distance from the center point by factor k.

3. Applications of Dilation

Details: Dilation is used in computer graphics, image processing, engineering design, and geometric transformations. It's fundamental for scaling objects while maintaining proportions.

4. Using the Calculator

Tips: Enter the center point coordinate, original point coordinate, and scale factor. Positive scale factors enlarge, while factors between 0 and 1 reduce the distance.

5. Frequently Asked Questions (FAQ)

Q1: What does a negative scale factor do?
A: A negative scale factor performs both scaling and reflection across the center point.

Q2: What happens when k = 1?
A: The point remains unchanged as the distance from the center isn't scaled.

Q3: How does this extend to 2D or 3D space?
A: Apply the same formula separately to each coordinate (x, y, z).

Q4: What's the difference between dilation and translation?
A: Dilation scales distance from a fixed point, while translation moves all points by a fixed amount.

Q5: Can this be used for vector scaling?
A: Yes, when the center point is at the origin (0), this becomes simple scalar multiplication.