Analyze point images under glide reflections instantly. Choose axes, custom lines, distances, precision, and directions. See every step before exporting clean tables and reports.
A glide reflection combines a reflection across a line with a translation along that same line.
Reflection: (x, y) → (x, 2k - y)
Glide: (x, y) → (x + g, y) or (x - g, y)
Final: (x', y') = (x ± g, 2k - y)
Reflection: (x, y) → (2h - x, y)
Glide: (x, y) → (x, y + g) or (x, y - g)
Final: (x', y') = (2h - x, y ± g)
d = (Ax + By + C) / (A² + B²)
Reflected point: xr = x - 2Ad, yr = y - 2Bd
Unit direction along the line: u = (-B, A) / √(A² + B²)
Glide vector: T = ±g · u
Final point: (x', y') = (xr, yr) + T
The calculator also writes the glide reflection as one matrix-and-shift rule:
[x'] [m11 m12][x] + [qx]
[y'] = [m21 m22][y] [qy]
| Case | Input Point | Axis or Line | Glide Distance | Reflected Point | Final Image |
|---|---|---|---|---|---|
| Horizontal | P(4, 3) | y = 1 | 5 | (4, -1) | (9, -1) |
| Vertical | A(-2, 6) | x = 3 | 4 | (8, 6) | (8, 10) |
| General | B(2, 5) | x - y = 0 | 4 | (5, 2) | (7.8284, 4.8284) |
In the third example, the positive glide direction follows vector (-B, A) = (1, 1).
A glide reflection is a reflection across a line followed by a translation along the same line. It is one of the standard plane isometries in geometry.
For a translation parallel to the reflection line, reflecting then gliding gives the same result as gliding then reflecting. This is why the calculator can present one combined affine rule.
For the line Ax + By + C = 0, vector (A, B) is normal to the line. Rotating that normal gives a direction along the line, which can be written as (-B, A).
Yes. The calculator accepts decimals for points, glide distance, and line values. You can also control the displayed precision from zero to ten decimal places.
If the point already lies on the line, the reflection keeps it fixed. The final image then comes only from the glide translation along that line.
Reverse direction changes the sign of the glide vector. In horizontal mode it moves left, in vertical mode it moves downward, and in general mode it follows vector (B, -A).
The exported CSV and PDF files help with homework checks, classroom demonstrations, worksheet creation, and proof verification. They also make it easier to store transformation results.
Yes. Reflection preserves perpendicular distance to the line, and the glide translation is parallel to that line. Together they keep lengths and angles unchanged.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.