appendix-4-on-the-change-of-coordinates

Appendix Chapter 4 — On the Change of Coordinates

Summary: The 3D analogue of Book II’s chapter 2. Builds up the most general rigid motion in space in three stages — origin translation (§87), in-plane rotation $CT$ of the $xy$-axis (§§88–89, figure 140), tilt of the $xy$-plane about $AP$ (§90, figure 141), and the full composition (§§91–92) — yielding a six-parameter substitution $x=x(p,q,r),y=y(p,q,r),z=z(p,q,r)$ that leaves the surface invariant. From this Euler extracts degree invariance (§§93–94), planes = first-order surfaces (§§96–97), and the angle of inclination of a given plane $\alpha x+\beta y+\gamma z=a$ to each coordinate plane (§§99–100, figure 142).

Sources: appendix4, §§86–100. Figures 140, 141 in figures138-141. Figure 142 in figures142-144.

Last updated: 2026-05-12.

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§§86–87 — Origin translation

Just as a plane curve admits infinitely many coordinate equivalents, a surface admits infinitely more — the cutting plane itself can vary. The simplest change: keep the axis direction, slide the origin. $x=t\pm a,y=u\pm b,z=v\pm c$ leaves the surface invariant in shape but introduces three free parameters.

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

Pick any line $CT$ in the plane $APQ$ to be the new abscissa axis. Drop $CB\perp AP$ with $AB=a,BC=b$, and let $\angle RCT=\zeta$. New coordinates $CT=t,TQ=u,QM=z$. Then

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

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

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

Now the plane of the first two coordinates changes. Keep $AP$ as axis; rotate the second plane about it by angle $\eta$. From any surface point $M$, drop $MT$ onto the new plane $APT$; new coordinates $AP=x,PT=u,TM=v$. Then

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

§§91–92 — Full rigid-motion composition

Combine §89 (in-plane rotation by $\zeta$) and §90 (out-of-plane tilt by $\eta$), then further rotate the new abscissa axis in the new plane by $\theta$ about the line $CT$, with displacements $AB=a,BC=b$ and reference angle $CTR=\theta$. Substituting:

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

Six arbitrary constants $a,b,\zeta ,\eta ,\theta$ and (through $f,g,h$) the three translations. See change-of-coordinates-3d.

§§93–96 — Degree invariance

Although the general equation is usually very complicated, still if we consider the total degree of the coordinates, that total degree is always equal to the total degree of the original coordinates $x,y,z$. (source: appendix4, §94)

The sphere $x^2+y^2+z^2=a^2$ is second-degree and stays second-degree (§94). An $n$th-order surface admits sections of degree $\le n$ (§95), recovering the §50 invariance of oblique-plane-section-method from a higher vantage point. See degree-invariance-surfaces.

§§96–98 — First-order surfaces are planes

Any surface whose every plane section is a straight line must itself be a plane — the analogue of the straight-line-equation theorem. From the most general equation (§98): a judicious choice of the six constants reduces $\alpha x+\beta y+\gamma z=a$ to $r=f$, manifestly a plane parallel to the $pq$-plane.

§§99–100 — Position of $\alpha x+\beta y+\gamma z=a$ (figure 142)

Setting $z=0$ gives the trace $BCR$ of the plane on the principal coordinate plane: $\tan (\angle ACB)=\alpha /\beta$. Building the right triangle $QSM$ at any surface point $M$ (with $QS\perp BC$ and $MS$ the projection of $QM$), one finds

$$\tan QSM=-\frac{\sqrt{{\alpha }^2+{\beta }^2}}{\gamma },\cos QSM=\frac{\gamma }{\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}}.$$

By cyclic symmetry the inclinations to the three coordinate planes have tangents

$$\frac{-\sqrt{{\alpha }^2+{\beta }^2}}{\gamma },\frac{-\sqrt{{\alpha }^2+{\gamma }^2}}{\beta },\frac{-\sqrt{{\beta }^2+{\gamma }^2}}{\alpha }.$$

See plane-inclination-angles.

Cross-references

• 3D lift of coordinate-transformations (Book II Chapter 2 §§17–24).
• §94 lifts degree-invariance from curves to surfaces.
• §§96–98 lift straight-line-equation.
• §§99–100 form the bridge to Appendix Chapter 5: the canonical reduction of every second-order surface to $A{p}^2+B{q}^2+C{r}^2+K=0$ depends on being able to align coordinate axes with the principal diametral planes.

Figures

Figures 138–141

Figures 142–144

Related pages

• change-of-coordinates-3d
• degree-invariance-surfaces
• plane-inclination-angles
• appendix-5-on-second-order-surfaces
• appendix-1-on-the-surfaces-of-solids
• oblique-plane-section-method
• chapter-2-on-the-change-of-coordinates