Enter a point and line equation to find its reflected mirror point
Point X0
Point Y0
Line A
Line B
Line C
Line: Ax + By + C = 0
Result
Foot of Perpendicular M
-
Reflected Point P
-
Detailed Derivation
Reflection Formula
d = (Ax0 + By0 + C) / (A^2 + B^2)
Foot M = (x0 - Ad, y0 - Bd)
Reflected P = (x0 - 2Ad, y0 - 2Bd)
Point reflection over a line creates a symmetric mirror image. The original point and its reflection are equidistant from the line, with the line as the perpendicular bisector of the segment connecting them.
⚠If the point lies exactly on the line (Ax0+By0+C=0), the reflected point equals the original point.
What Is Point Reflection Over a Line?
Point reflection over a line, also known as mirroring, finds the symmetric counterpart of a point across a given line. It is a fundamental transformation in geometry with applications in graphics, optics, and symmetry analysis.
Perpendicular Condition
The segment connecting P and P is perpendicular to the reflection line. Their slopes are negative reciprocals.
Midpoint Condition
The foot of perpendicular M is the midpoint of P and P. Both points are equidistant from the line.
Distance Equality
The distance from P to line L equals the distance from P to line L. This ensures the reflection is symmetric.
Involution
Reflection is an involution: reflecting the reflected point returns the original point. Applying reflection twice gives identity.
Teaching Example: Point P(3,4) over line x+y-6=0 (A=1,B=1,C=-6). d = (3+4-6)/(1+1) = 1/2=0.5. Foot M = (3-0.5,4-0.5) = (2.5,3.5). Reflected P = (3-1,4-1) = (2,3). Verify: midpoint of P and P = (2.5,3.5) = M.
Use the formula: P = (x0 - 2Ad, y0 - 2Bd) where d = (Ax0+By0+C)/(A^2+B^2). The foot M is the midpoint between P and P.
What if the point is on the line?▼
If the point lies exactly on the line (Ax0+By0+C=0), then d=0 and the reflected point equals the original point. The point is its own reflection.
What is the reflection of (x,y) over y=x?▼
Reflecting (x,y) over y=x gives (y,x) - simply swap the coordinates. Over y=-x gives (-y,-x) - swap and negate both.
How is reflection different from rotation?▼
Reflection creates a mirror image (flip) across a line. Rotation turns the point around a center. Reflection reverses orientation (handedness), while rotation preserves it.
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