C3 - Transformations in three dimensions

C3 - Transformations in three dimensions

A mirror plane changes one coordinate, while a rotation about an axis leaves that axis coordinate unchanged. You will represent a single reflection in a coordinate plane or a single rotation about a coordinate axis using a 3×33\times3 matrix. Choosing the unchanged coordinate and checking the turning direction makes the matrices easier to understand.

Single reflections and rotations in 3D

What must change when a point crosses a coordinate mirror plane? Take P(2,−1,3)P(2,-1,3). Its reflection in z=0z=0, the xyxy-plane, is (2,−1,−3)(2,-1,-3): the perpendicular coordinate changes sign, and the two coordinates along the plane stay the same.

In three dimensions, the columns of a transformation matrix are the images of e1=(1,0,0)T\mathbf{e}_1=(1,0,0)^{\mathsf T}, e2=(0,1,0)T\mathbf{e}_2=(0,1,0)^{\mathsf T} and e3=(0,0,1)T\mathbf{e}_3=(0,0,1)^{\mathsf T}. The matrix acts on the position column r=(x,y,z)T\mathbf{r}=(x,y,z)^{\mathsf T}. Reflection in z=0z=0 keeps the first two unit vectors and reverses the third, giving the diagonal entries 1,1,−11,1,-1.

Reflections in coordinate planes

Mirror planeImage of (x,y,z)Matrixx=0(−x,y,z)(−100010001)y=0(x,−y,z)(1000−10001)z=0(x,y,−z)(10001000−1)\begin{array}{c|c|c} \text{Mirror plane}&\text{Image of }(x,y,z)&\text{Matrix}\\ \hline x=0&( -x,y,z)&\begin{pmatrix}-1&0&0\\0&1&0\\0&0&1\end{pmatrix}\\[6pt] y=0&(x,-y,z)&\begin{pmatrix}1&0&0\\0&-1&0\\0&0&1\end{pmatrix}\\[6pt] z=0&(x,y,-z)&\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix} \end{array}

The coordinate-plane reflection matrices are not separately printed in the formula booklet: derive them by changing the perpendicular coordinate's sign.

Every point in the mirror plane remains fixed. Points off the plane lie at equal perpendicular distances on opposite sides. Each of these matrices has determinant −1-1: volume is preserved and orientation reversed.

Worked example 1: reflection in the horizontal coordinate plane

Question. Find the matrix of reflection in z=0z=0, and hence the image of P(2,−1,3)P(2,-1,3).

Plan. Keep the coordinates parallel to the plane and reverse the perpendicular coordinate.

Fz=(10001000−1),Fz(2−13)=(2−1−3)\mathbf{F}_z=\begin{pmatrix}1&0&0\\0&1&0\\0&0&-1\end{pmatrix},\qquad \mathbf{F}_z\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}2\\-1\\-3\end{pmatrix}

Answer. The matrix is Fz\mathbf{F}_z above; P′=(2,−1,−3)P'=(2,-1,-3).

Check. The midpoint (2,−1,0)(2,-1,0) lies in z=0z=0. The joining segment is parallel to the zz-axis and has endpoints at equal distance 3 from the plane.

Change only the mirror plane to y=0y=0. A different coordinate must now reverse; the point stays the same.

Worked example 2: changing the mirror plane

Question. Find the matrix of reflection in y=0y=0, and hence the image of P(2,−1,3)P(2,-1,3).

Plan. Reverse the second coordinate and keep the first and third.

Fy=(1000−10001),Fy(2−13)=(213)\mathbf{F}_y=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&1\end{pmatrix},\qquad \mathbf{F}_y\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}2\\1\\3\end{pmatrix}

Answer. The matrix is Fy\mathbf{F}_y above; P′=(2,1,3)P'=(2,1,3).

Check. The midpoint (2,0,3)(2,0,3) lies in y=0y=0, and the segment is parallel to its perpendicular yy-axis. Both endpoints are distance 1 from the plane.

Now turn around an axis instead of reflecting across a plane. The axis coordinate stays unchanged; the two perpendicular coordinates undergo an ordinary plane rotation.

We use the positive rotation convention: a positive angle is anticlockwise when viewed from the positive end of the named axis towards the origin. For a positive quarter-turn, the cyclic directions are

Axis of rotationDirection changexe2↦e3ye3↦e1ze1↦e2\begin{array}{c|c} \text{Axis of rotation}&\text{Direction change}\\ \hline x&\mathbf{e}_2\mapsto\mathbf{e}_3\\ y&\mathbf{e}_3\mapsto\mathbf{e}_1\\ z&\mathbf{e}_1\mapsto\mathbf{e}_2 \end{array}

Looking from the other end reverses the apparent clockwise/anticlockwise direction, so state the viewpoint when describing a turn.

Rotations about coordinate axes

With the positive convention above,

Rx(θ)=(1000cos⁡θ−sin⁡θ0sin⁡θcos⁡θ)\mathbf{R}_x(\theta)=\begin{pmatrix}1&0&0\\0&\cos\theta&-\sin\theta\\0&\sin\theta&\cos\theta\end{pmatrix} Ry(θ)=(cos⁡θ0sin⁡θ010−sin⁡θ0cos⁡θ)\mathbf{R}_y(\theta)=\begin{pmatrix}\cos\theta&0&\sin\theta\\0&1&0\\-\sin\theta&0&\cos\theta\end{pmatrix} Rz(θ)=(cos⁡θ−sin⁡θ0sin⁡θcos⁡θ0001)\mathbf{R}_z(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\\sin\theta&\cos\theta&0\\0&0&1\end{pmatrix}

These three matrices are in the formula booklet. Use a negative angle for the reverse rotation.

For Rx\mathbf{R}_x, the pair (y,z)(y,z) rotates; for Rz\mathbf{R}_z, the pair (x,y)(x,y) rotates. The yy-axis formula looks different because the positively ordered pair is (z,x)(z,x). In that order,

z′=zcos⁡θ−xsin⁡θ,x′=zsin⁡θ+xcos⁡θz'=z\cos\theta-x\sin\theta,\qquad x'=z\sin\theta+x\cos\theta

Writing the coordinates back in (x,y,z)(x,y,z) order produces the booklet matrix. Do not move the minus sign just to make it resemble the xx-axis matrix.

Worked example 3: a quarter-turn about the z-axis

Question. Find the matrix of rotation through π2\frac{\pi}{2} about the zz-axis in the positive convention, and the image of P(2,−1,3)P(2,-1,3).

Plan. Leave zz unchanged and rotate the (x,y)(x,y) pair, using cos⁡π2=0\cos\frac{\pi}{2}=0 and sin⁡π2=1\sin\frac{\pi}{2}=1.

Rz=(0−10100001),Rz(2−13)=(123)\mathbf{R}_z=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&1\end{pmatrix},\qquad \mathbf{R}_z\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}1\\2\\3\end{pmatrix}

Answer. The matrix is Rz\mathbf{R}_z above; P′=(1,2,3)P'=(1,2,3).

Check. The height stays 3. In the perpendicular plane, (2,−1)(2,-1) turns anticlockwise to (1,2)(1,2), with dot product 2(1)+(−1)(2)=02(1)+(-1)(2)=0 and unchanged squared radius 5.

Change only the rotation axis from zz to yy. Keep the point and positive quarter-turn angle unchanged.

Worked example 4: a quarter-turn about the y-axis

Question. Find the matrix of rotation through π2\frac{\pi}{2} about the yy-axis in the positive convention, and the image of P(2,−1,3)P(2,-1,3).

Plan. Use the yy-axis formula, which leaves the middle coordinate unchanged.

Ry=(001010−100),Ry(2−13)=(3−1−2)\mathbf{R}_y=\begin{pmatrix}0&0&1\\0&1&0\\-1&0&0\end{pmatrix},\qquad \mathbf{R}_y\begin{pmatrix}2\\-1\\3\end{pmatrix}=\begin{pmatrix}3\\-1\\-2\end{pmatrix}

Answer. The matrix is Ry\mathbf{R}_y above; P′=(3,−1,−2)P'=(3,-1,-2).

Check. The coordinate along the axis remains −1-1. The perpendicular pair (z,x)=(3,2)(z,x)=(3,2) becomes (−2,3)(-2,3), a positive quarter-turn with unchanged squared radius 13. In particular, positive zz turns towards positive xx, confirming the sign convention.

[DIAGRAM: asset_name: A 3D rotation becomes a plane rotation; asset_slug: aqa7367-coordinate-axis-rotation; caption: A positive quarter-turn about the y-axis rotates the ordered pair (z,x) and leaves y unchanged; description: A left coordinate diagram in the plane y=-1 shows axes labelled z horizontally and x vertically, with the perpendicular pair (3,2) rotating anticlockwise to (-2,3) along a quarter-circle about the origin of that plane. A right panel records the corresponding 3D coordinates P=(2,-1,3) and P prime=(3,-1,-2), highlights the unchanged y coordinate, and states that positive z turns towards positive x.]
A positive quarter-turn about the y-axis rotates the ordered pair (z,x) and leaves y unchanged

Rotations preserve lengths and have determinant +1+1. A coordinate-plane reflection reverses one basis direction; a half-turn about a coordinate axis reverses the other two. For example, diag⁡(1,−1,−1)\operatorname{diag}(1,-1,-1) is rotation through π\pi about the xx-axis, not reflection in x=0x=0. At angle zero, each rotation matrix is the identity; it need not identify a unique axis.

The matrices here each describe one reflection in x=0x=0, y=0y=0 or z=0z=0, or one rotation about a coordinate axis. Use the appropriate single matrix directly.

Watch out

A rotation about the yy-axis keeps yy fixed; reflection in the plane y=0y=0 reverses yy. An axis and a plane with the same coordinate letter play different roles.

Try the remaining coordinate axis with a reversed angle.

Quick Recap

  • Reflecting in a coordinate plane reverses its perpendicular coordinate.
  • Rotating about a coordinate axis keeps its axis coordinate and rotates the perpendicular pair.
  • For the booklet matrices, view a positive angle from the positive end of the axis towards the origin.
  • A single reflection reverses orientation; a single rotation preserves it.

The opening reflection in z=0z=0 changes exactly the third coordinate because that is the direction perpendicular to the mirror plane.