Intersection of Two Surfaces

Summary: §§137–138 of the Appendix on Surfaces. Given two surfaces $F_1(x,y,z)=0,F_2(x,y,z)=0$ in three coordinates, the intersection is found by eliminating one variable between the two equations. The result is an equation in the other two, giving the projection of the intersection onto the coordinate plane spanned by them. Special case: surface meets plane $\alpha z+\beta y+\gamma x=f$ — substitute $z=(f-\beta y-\gamma x)/\alpha$ to get the section equation, identical to the oblique-plane-section-method.

Sources: appendix6, §§137–138.

Last updated: 2026-05-12.

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§137 — Setup

Just as the intersection of two planes is a straight line, so the intersection of two surfaces is either a straight line or a curve, which may lie on a plane or it may not. Whatever kind it may be, each of its points must lie on both surfaces, so that it will satisfy both equations. (source: appendix6, §137)

So a point $(x,y,z)$ lies on the intersection iff it satisfies both surface equations simultaneously. Two equations in three variables yield a 1D locus.

§138 — Elimination procedure

Suppose we are given two surfaces which intersect each other, each expressed by an equation in three coordinates with respect to the same set of principal axes. Thus we have two equations in the three coordinates $x,y$ and $z$. If we eliminate one of these coordinates, we have an equation in the two other variables which gives the projection of the intersection onto the plane determined by these two coordinates. (source: appendix6, §138)

Eliminate $z$ (using a resultant or substitution) → equation $E(x,y)=0$, the projection onto plane $xy$. Eliminate $y$ → equation $E^{\prime }(x,z)=0$, the projection onto plane $xz$. Eliminate $x$ → equation $E^{\prime \prime }(y,z)=0$, the projection onto plane $yz$.

By non-planar-curves-by-projection §135, two of these projections together determine the space curve.

Surface ∩ plane as the simplest case

When one of the surfaces is the plane $\alpha z+\beta y+\gamma x=f$, the elimination is trivial:

$$z=\frac{f-\beta y-\gamma x}{\alpha }.$$

Substitute into the other surface equation $F(x,y,z)=0$ → equation $F(x,y,(f-\beta y-\gamma x)/\alpha )=0$ in $(x,y)$ only — the projection of the section onto the plane $xy$. This is precisely the oblique-plane-section-method §49 in algebraic form (with $\alpha ,\beta ,\gamma$ as the plane normal components).

Two-surface examples

The chapter cashes this out in several worked examples:

• Sphere meets plane (§§140–141): $x^2+y^2+z^2=a^2$ ∩ $\alpha z+\beta y+\gamma x=f$ gives the projection $$({\alpha }^2+{\beta }^2)y^2+2\beta \gamma xy+({\alpha }^2+{\gamma }^2)x^2-2\beta fy-2\gamma fx+f^2-{\alpha }^2{a}^2=0,$$ an ellipse in the $xy$-plane. (Modern note: this is the projection of the circle of intersection — itself a circle in space.)
• Cone meets sphere (§§143–146): see tangency-of-surfaces.

Cross-references

• Projections feed into the curve-reconstruction method of non-planar-curves-by-projection §§135–136.
• The single-projection method here is the linear lift of oblique-plane-section-method from one surface to two surfaces.
• When the second surface is a plane, the result is the section algorithm of appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane. Hence Chapter 6 generalizes Chapter 2’s method.

Related pages

• appendix-6-on-the-intersection-of-two-surfaces
• non-planar-curves-by-projection
• tangency-of-surfaces
• oblique-plane-section-method
• appendix-2-on-the-intersection-of-a-surface-and-an-arbitrary-plane
• bezout-bound-surfaces
• planar-intersection-condition