Point Reflection Calculator – Mirror a Point Across a Center
A point reflection (also known as an inversion through a point or a 180° rotation about a center) creates a mirror image of a point on the opposite side of a fixed center of reflection. This calculator computes the reflected coordinates in both 2D and 3D coordinate space, making it useful for coordinate geometry homework, computer graphics transformations, physics simulations, and symmetric layout design.
The Point Reflection Formula
For a point P = (x, y) reflected across a center C = (cx, cy), the reflected point P' is:
x' = 2·cx − xy' = 2·cy − y
This formula comes directly from the midpoint relationship: the center of reflection is always the midpoint between the original point and its reflected image. Since cx = (x + x') / 2, solving for x' gives x' = 2·cx − x. For example, reflecting the point (3, 5) across the center (1, 2) gives x' = 2(1) − 3 = −1 and y' = 2(2) − 5 = −1, so the reflected point is (−1, −1).
Reflecting Across the Origin
The most common case is reflecting a point across the origin (0, 0). Since cx = cy = 0, the formula simplifies to negating each coordinate: (x, y) → (−x, −y). For instance, reflecting (4, −3) across the origin produces (−4, 3).
Extending to Three Dimensions
In 3D space, the same relationship applies to the Z-axis:
z' = 2·cz − z
So reflecting (3, 5, 7) across the center (1, 2, 3)produces (−1, −1, −1) — each coordinate is mirrored independently through the corresponding coordinate of the center.
Distance From Center and Midpoint Verification
Alongside the reflected coordinates, the calculator reports the distance from the original point to the center, computed with the standard distance formula d = √((x − cx)² + (y − cy)²) (extended with a Z-term in 3D). It also performs a midpoint verification — recomputing ((x + x')/2, (y + y')/2) and confirming it equals the center — as a quick sanity check that the reflection was applied correctly.
Batch Reflection of Multiple Points
Many practical problems require reflecting several points across the same center at once — for example, mirroring the vertices of a triangle or polygon to create a symmetric counterpart. Batch mode accepts a list of points (one per line, in x,y format) and reflects each one across the same center, returning a full table of results that can be exported as CSV.
Common Applications
- Coordinate geometry education — teaching students how reflective symmetry and the midpoint relationship work together
- Computer graphics — mirroring shapes, sprites, or scene elements around a pivot point
- Physics simulations — modeling symmetric forces, fields, or particle positions around a central point
- Architectural and engineering design — laying out symmetric structures or components around a central reference point
- CAD and vector graphics workflows — computing mirrored control points for paths and curves
Precision and Accuracy
All calculations use standard 64-bit floating-point arithmetic, which is accurate to about 15 significant digits. Because reflection only involves multiplication and subtraction (no trigonometric functions), results are exact for integer inputs and are rounded to your chosen precision — from 0 to 10 decimal places — for decimal inputs.