Reflection Point Calculator

Calculator Inputs

Point x *

Point y *

Reflection type *

p for x = p *

q for y = q *

h (center x) *

k (center y) *

a (ax + by + c = 0) *

b (ax + by + c = 0) *

c (ax + by + c = 0) *

Reset

Formula Used

This tool reflects a point (x, y) using closed-form coordinate rules.

Common symmetry lines

Across x-axis: (x, y) → (x, −y)
Across y-axis: (x, y) → (−x, y)
Across origin: (x, y) → (−x, −y)
Across y = x: (x, y) → (y, x)
Across y = −x: (x, y) → (−y, −x)
Across x = p: x′ = 2p − x, y′ = y
Across y = q: x′ = x, y′ = 2q − y
Across point (h, k): x′ = 2h − x, y′ = 2k − y

Across an arbitrary line

For line ax + by + c = 0:

d = (ax + by + c) / (a² + b²)
x′ = x − 2ad
y′ = y − 2bd

This works for any non-degenerate line where a and b are not both zero.

How to Use This Calculator

Enter the input point coordinates x and y.
Select a reflection type from the dropdown.
If needed, fill the extra parameters (p, q, h, k, or a, b, c).
Press Compute Reflection to view the result above.
Use CSV or PDF buttons to export the latest result.

Example Data Table

Input (x, y)	Reflection type	Parameters	Reflected (x′, y′)
(3, -2)	Across x-axis	—	(3, 2)
(5, 1)	Across point (h, k)	(h, k) = (2, -1)	(-1, -3)
(2, 4)	Across line ax + by + c = 0	a = 1, b = -1, c = 0	(4, 2)

Tip: The last example uses the line x − y = 0, which is y = x.

FAQs

1) What is a reflection point result?

It is the mirror image of your input point across a chosen axis, line, or center point. Distances to the mirror are preserved, but the point’s orientation relative to that mirror flips.

2) When should I use the line form ax + by + c = 0?

Use it when the reflection line is not a simple axis or diagonal. It supports any straight line, including vertical lines. Just ensure a and b are not both zero.

3) Does reflecting across a point mean rotation?

Reflecting across a point (h, k) is equivalent to a 180° rotation around that point. The formulas x′ = 2h − x and y′ = 2k − y capture that transformation.

4) Why do I see Δx, Δy, and distance?

They quantify how far the reflected point moved from the original. This is useful for debugging and geometry checks, especially for arbitrary lines where visual intuition can be tricky.

5) Can this handle negative and decimal values?

Yes. Inputs accept integers, decimals, and negative values. The calculator formats outputs cleanly while preserving precision suitable for most classroom and engineering-style problems.

6) What happens if I enter a = 0 and b = 0?

That does not define a valid line, so the calculator will show an error. A line needs at least one of a or b to be nonzero for a meaningful reflection.

7) How do CSV and PDF exports work?

After you compute a result, the tool stores it as your latest calculation. The export buttons generate a downloadable file containing inputs, outputs, and step lines for recordkeeping.