change-of-coordinates-3d

Change of Coordinates in Three Dimensions

Summary: §§86–92 of the Appendix on Surfaces. The master rigid-motion substitution built up in three layers — in-plane rotation (§§88–89), out-of-plane tilt (§90), and full composition (§§91–92) — that maps original coordinates $(x,y,z)$ to new coordinates $(p,q,r)$ relative to an arbitrary new origin and orthonormal frame. Six free constants $a,b,\zeta ,\eta ,\theta ,f,g,h$ (translation triple absorbs $a,b$ into $f,g,h$). The 3D analogue of Book II’s chapter-2 coordinate-transformations.

Sources: appendix4, §§86–92. Figures 140, 141 in figures138-141.

Last updated: 2026-05-12.

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§87 — Translation only

If only the origin changes:

$$x=t\pm a,y=u\pm b,z=v\pm c.$$

The new coordinates $(t,u,v)$ are parallel to the old ones. Three free constants.

§§88–89 — Rotation of the abscissa axis in the same $xy$-plane (figure 140)

Pick a new axis $CT$ in the plane $APQ$. The triangle data: $AB=a,BC=b$, $\angle RCT=\zeta$, where $R$ is the foot of $CR$ parallel to the original axis $AP$.

For surface point $M$ with new coordinates $CT=t,TQ=u,QM=z$:

$$TR=t\sin \zeta ,CR=t\cos \zeta ,TS=u\sin \zeta ,QS=u\cos \zeta .$$

Substituting into the chain $AP=CR+TS-AB$ and $QP=QS-TR-BC$:

$$\boxed{x=t\cos \zeta +u\sin \zeta -a,y=u\cos \zeta -t\sin \zeta -b,z=z.}$$

Three free constants $a,b,\zeta$.

§90 — Tilt of the $xy$-plane about $AP$ (figure 141)

Now the second plane changes — rotate the plane $APQ$ about the axis $AP$ by angle $\eta =\angle QPT$. New coordinates $AP=x,PT=u,TM=v$ (with $MT\perp$ new plane).

Drop $TR\perp PQ$ and $TS\perp QM$:

$$TR=u\sin \eta ,PR=u\cos \eta ,TS=v\sin \eta ,MS=v\cos \eta .$$

Then $PQ=y=u\cos \eta -v\sin \eta$ and $QM=z=v\cos \eta +u\sin \eta$.

$$\boxed{x=x,y=u\cos \eta -v\sin \eta ,z=v\cos \eta +u\sin \eta .}$$

§91 — Both at once: rotated axis in a tilted plane

Combining §89 (with three constants) and §90 (with one new constant $\eta$). With $CT=p,TQ=q,QM=r$ in the new most-general system:

$$x=p\cos \zeta +q\sin \zeta -a,y=q\cos \zeta -p\sin \zeta -b,z=r.$$

But here $p,q$ are old §89 coordinates that themselves change under §90: $p=t,q=u\cos \eta -v\sin \eta ,r=v\cos \eta +u\sin \eta$.

Substitute:

$$\begin{array}{rl}x& =t\cos \zeta +u\sin \zeta \cos \eta -v\sin \zeta \sin \eta -a,\\ y& =-t\sin \zeta +u\cos \zeta \cos \eta -v\cos \zeta \sin \eta -b,\\ z& =u\sin \eta +v\cos \eta .\end{array}$$

§92 — Full composition: also rotate the axis in the new plane

One more rotation by $\theta$ in the new plane, with displacements $AB,BC$ giving constants $f,g,h$. Setting $CT=p,TQ=q,QM=r$ after this further rotation:

$$t=p\cos \theta +q\sin \theta -AB,u=-p\sin \theta +q\cos \theta -BC,v=r.$$

Substituting into §91 and collecting:

$$\begin{array}{rl}x& =p(\cos \zeta \cos \theta -\sin \zeta \cos \eta \sin \theta )+q(\cos \zeta \sin \theta +\sin \zeta \cos \eta \cos \theta )-r\sin \zeta \sin \eta +f,\\ y& =-p(\sin \zeta \cos \theta +\cos \zeta \cos \eta \sin \theta )-q(\sin \zeta \sin \theta -\cos \zeta \cos \eta \cos \theta )-r\cos \zeta \sin \eta +g,\\ z& =-p\sin \eta \sin \theta +q\sin \eta \cos \theta +r\cos \eta +h.\end{array}$$

Six arbitrary constants: $\zeta ,\eta ,\theta ,f,g,h$. (The original $a,b$ have been absorbed into $f,g$.)

§93 — Use of the most general equation

It is hardly possible to give an example in which it would be convenient to find the most general equation. The usefulness of the most general equation might be seen from the possibility of choosing the arbitrary constants in such a way that the equation would take its simplest form. (source: appendix4, §93)

In practice §92’s substitution is too unwieldy for direct calculation but invaluable as a theoretical instrument — it powers degree invariance (§94, see degree-invariance-surfaces), the canonical reduction of quadrics (Appendix Ch. 5 §§113–115, see quadric-canonical-form), and the proof that first-order surfaces are planes (§§97–98).

Cross-references

• Book II curve analogue: coordinate-transformations in chapter 2 §§17–24, with two angles instead of three.
• §92 supplies the substitution used in oblique-plane-section-method §49 (with the plane choice trivializing one angle).
• The angles $\zeta ,\eta ,\theta$ are Euler’s first appearance of Euler angles — the rotation parameterization of $SO(3)$.

Figures

Figures 138–141

Related pages

• appendix-4-on-the-change-of-coordinates
• coordinate-transformations
• degree-invariance-surfaces
• oblique-plane-section-method
• quadric-canonical-form