1. What is Dilation with Center?

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 by scaling it from a fixed point called the center of dilation.

2. How Does the Calculator Work?

The calculator uses the dilation formula:

Where:

• \( x' \) — Dilated point
• \( c_x \) — Center point of dilation
• \( k \) — Scale factor (k > 1 enlarges, 0 < k < 1 reduces)
• \( x \) — Original point

Explanation: The formula calculates the new position of a point after scaling it by factor k relative to the center point cₓ.

3. Importance of Dilation Calculations

Details: Dilation calculations are fundamental in computer graphics, image processing, geometric transformations, and scaling operations in various engineering and design applications.

4. Using the Calculator

Tips: Enter the center point, scale factor, and original point. The scale factor determines the type of dilation:

• k > 1: Enlargement
• 0 < k < 1: Reduction
• k = 1: No change
• k < 0: Includes reflection

5. Frequently Asked Questions (FAQ)

Q1: What happens when the scale factor is negative?
A: A negative scale factor produces a dilation combined with a reflection across the center point.

Q2: How does the center point affect the dilation?
A: The center point remains fixed during dilation. All other points move away from or toward this center depending on the scale factor.

Q3: Can this be extended to 2D or 3D coordinates?
A: Yes, the same formula applies independently to each coordinate (x, y, z) using the same center point and scale factor.

Q4: What's the difference between dilation and scaling?
A: Dilation is a specific type of scaling that uses a fixed center point, while general scaling might not preserve the position relative to a center.

Q5: How is this used in real-world applications?
A: Applications include image resizing, CAD designs, map scaling, and any situation requiring proportional resizing from a fixed point.