Advanced Reflect Point Calculator

Enter coordinates and choose any reflection rule confidently. Get exact transformed points, distances, and comparisons. Understand reflections clearly using visual plots and guided steps.

Calculator Input

Original x-coordinate

Original y-coordinate

Decimal places

Reflection type

Line value a for x = a

Line value b for y = b

Center h

Center k

Line coefficient a

Line coefficient b

Line coefficient c

The calculator supports axis reflection, diagonal reflection, fixed-line reflection, center reflection, and a full general-line method.

Plotly Graph

Example Data Table

Original Point	Reflection Rule	Result
(3, 5)	Across x-axis	(3, -5)
(4, -2)	Across y-axis	(-4, -2)
(6, 1)	Across y = x	(1, 6)
(1, 7)	Across x = 3	(5, 7)
(5, 2)	Through center (1, 1)	(-3, 0)
(4, 0)	Across x - y - 1 = 0	(1, 3)

Formula Used

The reflected point depends on the selected mirror rule.

Across x-axis: (x', y') = (x, -y)
Across y-axis: (x', y') = (-x, y)
Through origin: (x', y') = (-x, -y)
Across y = x: (x', y') = (y, x)
Across y = -x: (x', y') = (-y, -x)
Across x = a: (x', y') = (2a - x, y)
Across y = b: (x', y') = (x, 2b - y)
Through center (h, k): (x', y') = (2h - x, 2k - y)
Across ax + by + c = 0: x' = x - 2a(ax + by + c)/(a² + b²), y' = y - 2b(ax + by + c)/(a² + b²)

For every valid reflection, the midpoint between the original and reflected points lies on the mirror line or equals the reflection center.

How to Use This Calculator

Enter the original point coordinates.
Select the reflection type you need.
Fill any extra values such as line constants or center coordinates.
Choose the decimal precision for the displayed result.
Press Calculate Reflection to view the transformed point above the form.
Review the plotted graph, midpoint, movement distance, and formula used.
Use the CSV and PDF buttons to export the current result.

Frequently Asked Questions

1. What does a point reflection mean?

A point reflection creates a new point placed symmetrically across a mirror line or center. The original and reflected points stay the same distance from that mirror.

2. Can I reflect a point across any line?

Yes. Use the general line option and enter coefficients for ax + by + c = 0. The calculator applies the full coordinate geometry reflection formula automatically.

3. Why is the midpoint important?

The midpoint helps verify the result. For line reflections, that midpoint lies on the mirror line. For center reflections, the midpoint equals the reflection center.

4. What happens when I reflect across y = x?

The coordinates switch places. A point (x, y) becomes (y, x). This is one of the fastest and most common reflection transformations.

5. What happens when I reflect across y = -x?

The coordinates switch and both signs reverse. A point (x, y) becomes (-y, -x). This creates symmetry about the descending diagonal line.

6. Can this calculator handle decimal coordinates?

Yes. All inputs accept decimals, and you can control the number of displayed decimal places. That makes it useful for both classroom and technical work.

7. What does movement distance show?

Movement distance is the straight-line distance from the original point to the reflected point. It helps compare how far different reflection rules shift the same point.

8. When should I use the center reflection option?

Use center reflection when symmetry is defined around a single point instead of a line. It is equivalent to a 180-degree turn around that center.