tangent-plane-to-surface

Tangent Plane to a Surface

Summary: §§147–149 of the Appendix on Surfaces. Construction of the tangent plane at a point $M$ on a surface (figure 149). Cut the surface with two perpendicular planes through $M$ — one perpendicular to $APQ$ along $QS$ parallel to $AP$, the other perpendicular along $QP$. Each gives a plane curve whose tangent at $M$ can be computed by Book II’s tangent-by-translation method, with subtangents $QS=s,QT=t$. The tangent plane is the plane $SMT$ spanned by these two tangents. From the construction Euler reads off the trace direction $\tan AXS=t/s$, the inclination $\tan (MRQ)=z\sqrt{s^2+t^2}/(st)$, and the normal direction.

Sources: appendix6, §§147–149. Figure 149 in figures148-149.

Last updated: 2026-05-12.

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§147 — Heuristic

There is a much simpler way of finding the tangent plane for any surface. We use the method of finding tangents to curves… we suppose that the nature of the surface, whose tangent plane we are seeking, is given by an equation in the three coordinates $AP=x,PQ=y$, and $QM=z$. (source: appendix6, §147)

The key idea: a tangent plane is determined by any two tangent lines through the point. Take two plane sections through $M$ (using two different cutting planes that intersect in some line through $M$), compute the tangent to each section at $M$ using the plane-curve method tangent-by-translation from Book II Chapter 13, and the tangent plane is spanned by these two tangents.

§148 — Construction (figure 149)

Setup: surface point $M=(x,y,z)$. Drop perpendicular $MQ\perp APQ$.

First cutting plane: perpendicular to $APQ$, containing the line $QS$ in $APQ$ parallel to the axis $AP$. The section of the surface by this plane is a plane curve $EM$ in the cutting plane; its tangent at $M$ is the line $MS$, meeting the line $QS$ at $S$. The intercept $QS=s$ is the subtangent of this section.

Second cutting plane: perpendicular to $APQ$, containing the line $QP$ (the ordinate line). Section is the plane curve $FM$; tangent at $M$ is $MT$ with $QT=t$ the subtangent.

The tangent plane is the plane $SMT$ (the plane through the three points $S,M,T$).

§149 — Reading off slopes, inclinations, and normals

From the two subtangents:

$$PT=t-y,PX=s-sy/t,$$

where $X$ is the intersection of the tangent-plane trace $ST$ with the axis $AP$ in the plane $APQ$.

Slope of the tangent plane’s trace in $APQ$:

$$\tan (\angle AXS)=\frac{t}{s}.$$

Length $ST$:

$$ST=\sqrt{s^2+t^2}.$$

Inclination of the tangent plane to $APQ$: dropping perpendicular $QR$ from $Q$ to $ST$,

$$QR=\frac{st}{\sqrt{s^2+t^2}}.$$

Then $\tan (\angle MRQ)=QM/QR$, so

$$\boxed{\tan (\angle MRQ)=\frac{z\sqrt{s^2+t^2}}{st}.}$$

Normal to the tangent plane at $M$: drop $MN\perp$ tangent plane. Then $N$ lies in the plane $APQ$ at the foot of the perpendicular from $M$, with

$$PV=\frac{z^2}{s},NW=\frac{z^2}{t},$$

giving its position $N$ in the plane $APQ$. The straight line $MN$ is normal to both the surface and the tangent plane at $M$.

Modern reading: gradient as normal

In modern vector notation, the surface is $F(x,y,z)=0$. The tangent plane at $M$ has normal $\nabla F=(F_x,F_y,F_z)$. The subtangents $s,t$ correspond to the partial derivatives:

$$s=-\frac{y\cdot F_x}{F_z}\text{(approximately, with signs depending on figure orientation)}.$$

Euler’s two-subtangent construction is the implicit-function-theorem computation of the gradient, two centuries early.

Cross-references

• 3D lift of tangent-by-translation (Book II §§286–291) — that method substitutes $x\mapsto p+t,y\mapsto q+u$ into a plane curve and truncates to linear order; here it is applied twice (once for each cutting plane) to extract a tangent plane.
• The tangent plane construction is dual to the tangency-of-surfaces perspective: there, two surfaces tangent at a point share their tangent plane; here, a tangent plane is constructed for a single surface at a single point.
• The normal direction $MN$ is the analogue of the normal line in Book II subtangent-and-subnormal.

Figures

Figures 148–149

Related pages

• appendix-6-on-the-intersection-of-two-surfaces
• tangency-of-surfaces
• tangent-by-translation
• subtangent-and-subnormal
• intersection-of-two-surfaces
• plane-inclination-angles