Rotation Transformation Calculator – Rotate Points & Shapes
A rotation is a rigid transformation that turns a point or a whole figure around a fixed
center by an angle \(\theta\). Rotations preserve distances, angles,
perimeter, and area.
1) Angle direction and units
-
By convention, \(\theta > 0\) means counterclockwise (CCW) rotation.
-
\(\theta < 0\) means clockwise (CW) rotation.
-
Degrees and radians are related by \(\;\theta_{\text{rad}}=\theta_{\circ}\,\pi/180\).
2) Rotation about the origin
For a point \(P=(x,y)\), a rotation by angle \(\theta\) about the origin produces \(P'=(x',y')\) where:
\[
\begin{aligned}
x' &= x\cos\theta - y\sin\theta,\\
y' &= x\sin\theta + y\cos\theta.
\end{aligned}
\]
This can be written as a matrix multiplication:
\[
\begin{bmatrix}x'\\y'\end{bmatrix}
=
\begin{bmatrix}
\cos\theta & -\sin\theta\\
\sin\theta & \cos\theta
\end{bmatrix}
\begin{bmatrix}x\\y\end{bmatrix}.
\]
3) Rotation about a general point \((h,k)\)
To rotate around a center \(C=(h,k)\), use a translate–rotate–translate idea:
- Move the center to the origin: \(X=x-h,\; Y=y-k\).
- Rotate \((X,Y)\) about the origin.
- Translate back: add \((h,k)\).
The final formulas are:
\[
\begin{aligned}
x' &= h + (x-h)\cos\theta - (y-k)\sin\theta,\\
y' &= k + (x-h)\sin\theta + (y-k)\cos\theta.
\end{aligned}
\]
4) What rotations preserve (invariants)
-
Distance from the center: \(\;|CP'| = |CP|\) for any point \(P\).
-
Distance between points: for any two points \(A,B\),
\(\;|A'B'|=|AB|\).
-
Angles: the angle between any two segments is unchanged.
-
Area and perimeter: polygons keep the same area and perimeter.
5) Interpreting the diagram in this tool
-
The graph uses square units, so a unit step on the \(x\)-axis matches a unit step on the \(y\)-axis.
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You can pan by dragging and zoom with wheel/trackpad or pinch.
-
The Play button animates intermediate positions, making the rotation feel like a “clock hand” sweep.
-
If the first vertex lies extremely close to the center, the rotation arc may be hidden (the radius would be near zero).
6) Quick example
Rotate \(P=(1,0)\) by \(90^\circ\) CCW about the origin:
\[
\begin{aligned}
x' &= 1\cdot\cos 90^\circ - 0\cdot\sin 90^\circ = 0,\\
y' &= 1\cdot\sin 90^\circ + 0\cdot\cos 90^\circ = 1.
\end{aligned}
\]
So \(P'=(0,1)\).
7) Common mistakes
-
Mixing degrees and radians (a frequent source of wrong plots).
-
Forgetting that clockwise rotation is represented by a negative angle.
-
Rotating about \((h,k)\) without translating first (the origin formulas only work about \((0,0)\)).
