1. What is the Dilation Equation?

The dilation equation \( x' = k \times x \) describes a scaling transformation where an original value \( x \) is multiplied by a scale factor \( k \) to produce a dilated value \( x' \). This is fundamental in geometry, physics, and engineering applications.

2. How Does the Calculator Work?

The calculator uses the simple dilation equation:

Where:

• \( x' \) — Dilated value (units)
• \( k \) — Scale factor (unitless)
• \( x \) — Original value (units)

Explanation: The equation scales the original value by the factor \( k \). When \( k > 1 \), it's an enlargement; when \( 0 < k < 1 \), it's a reduction; when \( k < 0 \), it includes a reflection.

3. Applications of Dilation

Details: Dilation is used in image processing, computer graphics, map scaling, engineering drawings, and physics transformations. It's fundamental to understanding similarity in geometry.

4. Using the Calculator

Tips: Enter the scale factor \( k \) (can be positive or negative) and the original value \( x \). The calculator will compute the dilated value \( x' \).

5. Frequently Asked Questions (FAQ)

Q1: What does a negative scale factor mean?
A: A negative scale factor performs both scaling and reflection across the origin or axis.

Q2: How is this different from translation?
A: Dilation changes size while maintaining shape proportions; translation moves an object without changing its size or shape.

Q3: Can this be used for 2D or 3D scaling?
A: This calculator handles 1D scaling. For 2D/3D, you would need separate scale factors for each dimension.

Q4: What's the difference between uniform and non-uniform scaling?
A: Uniform scaling uses the same factor for all dimensions (as in this calculator), while non-uniform scaling uses different factors.

Q5: How does this relate to similarity transformations?
A: Dilation is a key component of similarity transformations, which preserve angles and proportional distances.