Point symmetry (also called central symmetry) means a shape looks the same after a
180° rotation about some center point \(C\).
It’s the same idea as saying “every part has an opposite part the same distance away on the other side of the center.”
Definition
A shape has point symmetry about a center \(C=(x_c,y_c)\) if for every point \(P=(x,y)\) in the shape,
the point \(P'\) defined by
Equivalently, \(C\) is the midpoint of the segment \(\overline{PP'}\):
How the calculator detects point symmetry
The calculator supports two interpretations:
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Polygon mode (ordered vertices): edges matter, so the vertex order must be consistent around the polygon.
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Point-set mode (unordered): only the set of points matters (useful for sampled curves, “Z vs S” point clouds, etc.).
Step 1 — Choose the center \(C\)
In Auto center mode, the calculator uses the average of all input points:
If the shape is centrally symmetric, this average equals the true center of symmetry.
You can also select Manual mode and input a center \(C\) to test symmetry about that specific point.
Step 2 — Check for opposite partners
For each point \(P_i\), compute the required partner \(P_i' = 2C - P_i\). Then:
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In Point-set mode, the calculator checks that the multiset of points equals the multiset of partners
(within a tolerance).
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In Polygon mode, a necessary condition is that the polygon has an even number of vertices \(n\).
Then the opposite vertex should be exactly \(n/2\) steps ahead in the vertex order:
\[
V_{i+n/2} \approx 2C - V_i.
\]
Tolerance
Real input often has rounding. The calculator uses a tolerance \(\varepsilon\) (scaled to your shape size) and treats
two points as matching when their distance is \(\le \varepsilon\).
If you paste approximate decimals, try a looser tolerance.
Examples and intuition
Parallelograms: YES
Every parallelogram is centrally symmetric: the center is the intersection of the diagonals.
Rectangles, rhombi, and squares are special cases of parallelograms, so they are also centrally symmetric.
Isosceles trapezoids: usually NO (but can have line symmetry)
An isosceles trapezoid typically has a line of symmetry, but not point symmetry.
Rotating it 180° does not map it onto itself (unless it collapses into a parallelogram).
“Z-shape” vs “S-shape”
A “Z-shape” is often point-symmetric: rotate it 180° about its center and it looks the same.
A typical “S-shape” is not point-symmetric (it may have other visual patterns, but not the strict 180° invariance).
If you sample points along a curve and paste them in, use Point-set mode.
Common issues
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Wrong polygon order: In polygon mode, vertices must be listed in order around the boundary.
If you are unsure, switch to point-set mode.
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Odd number of vertices: A centrally symmetric polygon must have an even number of vertices.
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Too strict tolerance: If your points are approximate decimals, a stricter tolerance may incorrectly fail.
Key takeaway
Point symmetry is a clean, testable condition:
pick a candidate center \(C\), reflect every point through \(C\), and check that the shape matches itself.
That’s exactly what this calculator visualizes with the center marker and the 180° overlay.
