Dilation Calculator

Geometric Dilation Calculator

Use this calculator to find the coordinates of a point after a geometric dilation. A dilation is a transformation that changes the size of a figure without changing its shape. It requires an original point, a center of dilation, and a scale factor.

function calculateDilation() { var originalX = parseFloat(document.getElementById('originalX').value); var originalY = parseFloat(document.getElementById('originalY').value); var centerX = parseFloat(document.getElementById('centerX').value); var centerY = parseFloat(document.getElementById('centerY').value); var scaleFactor = parseFloat(document.getElementById('scaleFactor').value); var resultDiv = document.getElementById('dilationResult'); if (isNaN(originalX) || isNaN(originalY) || isNaN(centerX) || isNaN(centerY) || isNaN(scaleFactor)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; return; } // Formula for dilation: // x' = cx + k * (x – cx) // y' = cy + k * (y – cy) var dilatedX = centerX + scaleFactor * (originalX – centerX); var dilatedY = centerY + scaleFactor * (originalY – centerY); resultDiv.innerHTML = 'The dilated point (X\', Y\') is: (' + dilatedX.toFixed(2) + ', ' + dilatedY.toFixed(2) + ')'; }

Understanding Geometric Dilation

Geometric dilation is a fundamental transformation in geometry that alters the size of a figure without changing its shape. It's like zooming in or out on an image. Every dilation requires two key pieces of information: a center of dilation and a scale factor.

What is a Center of Dilation?

The center of dilation is a fixed point in the plane from which all distances are measured. It's the "anchor" point around which the figure expands or contracts. If the center of dilation is (0,0) (the origin), the calculation simplifies, but it can be any point in the coordinate plane.

What is a Scale Factor?

The scale factor, often denoted by 'k', determines how much the figure is enlarged or reduced. It's a ratio that describes the size change.

• If k > 1, the dilation is an enlargement (the figure gets bigger).
• If 0 < k < 1, the dilation is a reduction (the figure gets smaller).
• If k = 1, the dilation results in a figure congruent to the original (no change in size).
• If k < 0, the dilation involves both a size change and a rotation of 180 degrees around the center of dilation.

How Dilation Works (The Formula)

To find the coordinates of a dilated point (X', Y') from an original point (X, Y) with a center of dilation (Cx, Cy) and a scale factor (k), we use the following formulas:

X' = Cx + k * (X – Cx)

Y' = Cy + k * (Y – Cy)

This formula essentially calculates the distance from the center of dilation to the original point, multiplies that distance by the scale factor, and then adds it back to the center of dilation's coordinates to find the new position.

Examples of Dilation

Let's look at a few examples to illustrate how dilation works:

Example 1: Enlargement from the Origin
Original Point (X, Y): (2, 3)
Center of Dilation (Cx, Cy): (0, 0)
Scale Factor (k): 2

Using the formulas:
X' = 0 + 2 * (2 – 0) = 0 + 2 * 2 = 4
Y' = 0 + 2 * (3 – 0) = 0 + 2 * 3 = 6
Dilated Point: (4, 6)

As you can see, the point moved further away from the origin, doubling its distance, resulting in an enlargement.

Example 2: Reduction from a Non-Origin Center
Original Point (X, Y): (5, 7)
Center of Dilation (Cx, Cy): (1, 1)
Scale Factor (k): 0.5

Using the formulas:
X' = 1 + 0.5 * (5 – 1) = 1 + 0.5 * 4 = 1 + 2 = 3
Y' = 1 + 0.5 * (7 – 1) = 1 + 0.5 * 6 = 1 + 3 = 4
Dilated Point: (3, 4)

Here, the point moved closer to the center of dilation (1,1), reducing its distance by half.

Example 3: Negative Scale Factor
Original Point (X, Y): (2, 4)
Center of Dilation (Cx, Cy): (0, 0)
Scale Factor (k): -1

Using the formulas:
X' = 0 + (-1) * (2 – 0) = 0 – 2 = -2
Y' = 0 + (-1) * (4 – 0) = 0 – 4 = -4
Dilated Point: (-2, -4)

A negative scale factor not only changes the size (if |k| is not 1) but also rotates the point 180 degrees around the center of dilation.

Applications of Dilation

Geometric dilation is not just a theoretical concept; it has numerous real-world applications:

• Maps and Cartography: Scaling maps up or down to represent different areas or levels of detail.
• Photography and Image Editing: Resizing images, zooming in or out, and creating special effects.
• Computer Graphics: Scaling objects in 2D and 3D environments for games, animations, and CAD software.
• Architecture and Engineering: Creating scaled models or blueprints of buildings and structures.
• Art and Design: Artists use dilation principles to create perspective and proportion in their work.

This calculator simplifies the process of finding dilated coordinates, making it easier to understand and apply the principles of geometric dilation.