Surface Coordinates and Equation

Summary: §§3–12 of the Appendix on Surfaces. Three mutually perpendicular axes meeting at $A$ assign each point $M$ in space three coordinates $AP=x$, $PQ=y$, $QM=z$ (figure 119). A surface is then given by an equation in these three variables, just as a plane curve is given by an equation in $(x,y)$. The role-switch among the three coordinates is the surface-analogue of oblique-coordinates.

Sources: appendix1, §§3–12. Figure 119 in figures119-120.

Last updated: 2026-05-12.

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The three-coordinate apparatus (§§4–5)

A table top represents a fixed plane. On it lies a directrix $AB$ with origin $A$. From any point $M$ of the surface drop the perpendicular $MQ$ to the table; from $Q$ drop the perpendicular $QP$ to the line $AB$. Then $AP,PQ,QM$ are three mutually perpendicular segments giving the three coordinates of $M$ (source: appendix1, §4).

Letting $AP=x,PQ=y,QM=z$, the nature of the surface is expressed by an equation in $x,y,z$ in which $z$ is given as a function of $x$ and $y$ (source: appendix1, §5):

• $z>0$: $M$ above the table;
• $z<0$: $M$ below;
• $z=0$: $M$ on the table;
• $z$ complex: no point of the surface on that perpendicular;
• multiple real $z$ at one $(x,y)$: the perpendicular meets the surface in several points.

Continuous vs. irregular surfaces (§6)

A continuous (regular) surface is one whose entire shape is captured by a single equation $z=f(x,y)$. An irregular surface is patched from several — e.g. a sphere joined to a cone. Euler restricts attention to the continuous case (source: appendix1, §6).

Single-valued and multi-valued (§§8–9)

If $z$ equals a non-irrational function $P(x,y)$, every perpendicular through $Q$ meets the surface in a single real point — never complex (source: appendix1, §8). The next levels are quadratic and cubic in $z$:

$$z^2-Pz+Q=0,z^3-P{z}^2+Qz-R=0,$$

giving two- and three-valued surfaces by exact analogy with the plane-curve case in multi-valued-curves (source: appendix1, §9). The intersection count of a vertical line with the surface is read directly from the number of real roots of $z$.

The single-valued/multi-valued distinction is frame-dependent: rotating the reference plane $APQ$ may turn a single-valued surface into a multi-valued one and vice versa (source: appendix1, §8). What matters for intrinsic study is the equation up to coordinate change.

Six coordinate orderings (§10)

Just as for plane curves the two coordinates $x,y$ can be swapped, here the three coordinates $x,y,z$ can be permuted in $3!=6$ ways. With three reference planes $APQp,APq\pi ,Ap\xi \pi$ and two axes per plane, there are six relations between the coordinates (source: appendix1, §10):

Plane	Choice 1	Choice 2
$APQp$	$AP=x,PQ=y,QM=z$	$Ap=y,pQ=x,QM=z$
$APq\pi$	$AP=x,Pq=z,qM=y$	$A\pi =z,\pi q=x,qM=y$
$Ap\xi \pi$	$Ap=y,p\xi =z,\xi M=x$	$A\pi =z,\pi \xi =y,\xi M=x$

In every case, the straight-line distance from $A$ to $M$ is

$$AM=\sqrt{x^2+y^2+z^2}.$$

The three-plane reference frame (§§11–12)

The same equation in $x,y,z$ describes the surface’s relation to all three reference planes (source: appendix1, §11): $z$ measures distance to $APQ$, $y$ to $APq$, $x$ to $Ap\xi$. The three coordinates $MQ=z,Mq=y,M\xi =x$ are pairwise perpendicular, locating $M$ as the corner of a parallelepiped opposite $A$ (figure 119, source: appendix1, §12). Sign conventions follow the orientation of each axis.

Cross-references

• Plane analogue: abscissa-and-ordinate (Book II Chapter 1).
• Multi-valued levels: multi-valued-curves.
• Six-fold permutations as a generalization of the oblique-coordinates axis choice.
• Worked example with a specific surface: §28 sphere $x^2+y^2+z^2=a^2$ in sections-by-coordinate-planes.

Figures

Figures 119–120

Related pages

• appendix-1-on-the-surfaces-of-solids
• algebraic-and-transcendental-surfaces
• diametral-plane
• octants-and-region-symmetries