Geometric Transformation Calculator

Results

Images of the points

Transformation matrix

Each step is a 3 × 3 matrix acting on (x, y, 1). The matrices multiply right to left, so the first step is the rightmost factor.

Before and after

Blue is the original, orange the final image, dotted outlines the shape after each earlier step, and green marks a reflection line or a center.

How to Use the Geometric Transformation Calculator

1. 1. Enter the points. Type one x, y pair per line, up to twelve. Three or more points are drawn as a polygon in the order you list them.
2. 2. Choose a transformation. Each step is a translation, reflection, rotation or dilation, with its own settings: the shift, the mirror line, the angle and center, or the scale factor and center.
3. 3. Chain steps if you need to. Add up to four steps; they are applied in the order shown. The dotted outlines in the plot show the shape after each earlier step.
4. 4. Read the images, the matrices and the plot. The table lists every point and its image; the matrices show the transformation as numbers; and the plot overlays the original on the result.

The Four Rigid and Similarity Transformations

Transformation Matrices

Translations are not linear, so a 2 × 2 matrix cannot describe them. The trick is homogeneous coordinates: write the point as (x, y, 1) and use 3 × 3 matrices. A translation by (a, b) becomes the matrix with rows (1, 0, a), (0, 1, b), (0, 0, 1); a rotation by θ becomes (cos θ, −sin θ, 0), (sin θ, cos θ, 0), (0, 0, 1); a dilation by k is (k, 0, 0), (0, k, 0), (0, 0, 1). Because every step is a matrix, a whole sequence is one matrix: the product of the step matrices, with the first step as the rightmost factor. A rotation about a center (c, d) is T(c, d) · R · T(−c, −d).

The determinant of the matrix's 2 × 2 corner tells you how areas change and whether the shape is mirrored. A determinant of 1 keeps area and orientation (translations and rotations); −1 keeps area but flips orientation (a single reflection); and k² scales areas by the square of the scale factor. The calculator reports that determinant along with the factor by which distances are multiplied.

Worked Examples

Start with the triangle A(1, 1), B(4, 1), C(1, 3), whose area is 3.

• Rotate 90° counterclockwise about the origin. (x, y) becomes (−y, x), so A′(−1, 1), B′(−1, 4), C′(−3, 1). The matrix is (0, −1, 0), (1, 0, 0), (0, 0, 1), with determinant 1.
• Reflect across the y-axis. (x, y) becomes (−x, y): A′(−1, 1), B′(−4, 1), C′(−1, 3). The determinant is −1, so the orientation is reversed.
• Translate by (3, −2). A′(4, −1), B′(7, −1), C′(4, 1).
• Dilate by k = 2 about the origin. A′(2, 2), B′(8, 2), C′(2, 6). Lengths double, and the area goes from 3 to 12 = 3 × 2².
• Rotate 45° about the origin. The point (1, 0) goes to (√2/2, √2/2), kept exact.
• Order matters. Take the point (1, 0). Rotate 90° and then translate by (1, 0): (1, 0) → (0, 1) → (1, 1). Translate first and then rotate: (1, 0) → (2, 0) → (0, 2).

Reflecting in Other Lines

For the line y = mx + b the calculator uses the reflection matrix about y = mx, which is (1 − m², 2m; 2m, m² − 1) divided by 1 + m², wrapped in translations by b. All entries are rational, so the result is exact. A vertical line x = c sends x to 2c − x. For example, reflecting the origin in y = x + 2 gives (−2, 2), because the line is the perpendicular bisector of the segment joining the two.

Frequently Asked Questions

What is a geometric transformation?

A rule that moves every point of the plane to a new position. Translations slide, reflections flip across a line, rotations turn about a center and dilations resize about a center. The first three are rigid motions that keep distances unchanged; a dilation keeps the shape but changes the size.

How do you rotate a point 90 degrees about the origin?

Counterclockwise, (x, y) becomes (−y, x); clockwise, it becomes (y, −x). For (3, 2) that gives (−2, 3) counterclockwise and (2, −3) clockwise. A 180° rotation sends (x, y) to (−x, −y).

How do you rotate about a point other than the origin?

Subtract the center from the point, rotate the result about the origin, then add the center back. The calculator does this for you through the matrix T(c) · R · T(−c), and also for dilations about a chosen center.

What happens to area under a dilation?

Lengths are multiplied by |k| and areas by k². A scale factor of 3 makes the shape three times as wide and nine times the area. A negative k also flips the shape through the center, which is the same as a dilation by |k| followed by a 180° rotation.

Does the order of transformations matter?

Usually yes. Matrix multiplication is not commutative, so a rotation followed by a translation differs from the translation followed by the rotation, as in the example above. Two rotations about the same center, or two translations, do commute.

How do I reflect across the line y = x?

Swap the coordinates: (x, y) becomes (y, x). Reflecting across y = −x sends (x, y) to (−y, −x). For any other line y = mx + b, choose that option in the reflection step and enter m and b.