appendix-6-on-the-intersection-of-two-surfaces

Appendix Chapter 6 — On the Intersection of Two Surfaces

Summary: Closes the Appendix and Book II. Non-planar curves in space need three coordinates, expressed by two equations in $(x,y,z)$ — two surfaces whose intersection is the curve. Eliminating one variable yields the projection onto a coordinate plane (figure 148); two projections determine the curve. Tangency is read off when the projection eliminant has a double root: worked examples include sphere-plane tangency $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$ (§141) and cone-sphere tangency along a circle (§§143–146). The tangent plane at a surface point $M$ is built from tangents to two perpendicular plane sections through $M$ (§§147–149, figure 149). The Bezout-type bound: two surfaces of orders $m,n$ have intersection projection of order $\le mn$ (§150). Closing example (§152): when does the intersection lie in a plane?

Sources: appendix6, §§131–152. Figures 148, 149 in figures148-149.

Last updated: 2026-05-12.

---

§§131–133 — Non-planar curves need two equations

A single equation in $(x,y,z)$ determines a surface, not a curve. Specifying a curve in space requires two equations in three variables; the third equation (in the remaining two) is implied by elimination.

The nature of any non-planar curve is most conveniently expressed by two equations in three variables. (source: appendix6, §133)

§§134–136 — Projection onto a coordinate plane (figure 148)

Given a non-planar curve $GMH$, drop from each point $M$ a perpendicular $MQ$ to the plane $BAC$. The feet $Q$ trace the projection curve $EQF$ in the plane $BAC$ whose equation is obtained by eliminating $z$ from the two given equations. Two projections (say onto $BAC$ and $BAD$) together determine the original curve $GMH$ in space. See non-planar-curves-by-projection.

§§137–138 — Intersection of two surfaces by elimination

Given two surfaces by equations in $(x,y,z)$, the intersection is the curve satisfying both. Eliminate one variable (e.g., $z$) → an equation in $(x,y)$ giving the projection onto the $xy$-plane. For the special case of a surface $F(x,y,z)=0$ cut by a plane $\alpha z+\beta y+\gamma x=f$, substitute $z=(f-\beta y-\gamma x)/\alpha$ — directly the oblique-plane-section-method in disguise. See intersection-of-two-surfaces.

§§139–146 — Tangency by double roots

A tangency is nothing but the coincidence of two intersections. (source: appendix6, §139)

If the projection equation has a double root (a perfect-square factor), the two surfaces touch along the corresponding subset:

• Single tangent point: the projection collapses to one point with two equal roots.
• Tangency along a curve: the projection equation has two equal nonlinear factors (e.g., divisible by a perfect square).
• Tangency in two points: four complex factors.

§§140–141 work the sphere-plane tangency: sphere $z^2+y^2+x^2=a^2$ cut by $\alpha z+\beta y+\gamma x=f$ projects to

$$f^2-{\alpha }^2{a}^2-2\beta fy-2\gamma fx+({\alpha }^2+{\beta }^2)y^2+2\beta \gamma xy+({\alpha }^2+{\gamma }^2)x^2=0,$$

an ellipse when the plane crosses the sphere, a single point (tangency) when $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$, with tangent point

$$(z,y,x)=\frac{1}{\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}}(\gamma a,\beta a,\alpha a).$$

§§143–146 work the cone-sphere tangency in detail: sphere $z^2+y^2+x^2=a^2$ and cone $(f-z{)}^2=m{x}^2+n{y}^2$ with vertex distance $f$ from the sphere’s center. For the right cone ($m=n$), tangency along a circle occurs when $f=a\sqrt{1+n}$. For the scalene cone, two isolated tangent points result. See tangency-of-surfaces.

§§147–149 — Tangent plane to a surface at $M$ (figure 149)

The construction:

1. Cut the surface at $M$ with the plane perpendicular to $APQ$ along the line $QS$ parallel to $AP$. Call this section $EM$. Its tangent at $M$, the line $MS$, hits $QS$ at $S$; $QS=s$ is the subtangent.
2. Cut again at $M$ with the plane perpendicular to $APQ$ along $QP$. Call this section $FM$. Tangent $MT$ has $QT=t$.
3. The tangent plane $SMT$ contains both tangents.

Reading off the orientation: $\tan AXS=t/s$ gives the trace’s slope in the $APQ$ plane; $\tan (MRQ)=z\sqrt{s^2+t^2}/(st)$ gives the plane’s inclination to $APQ$. The normal direction:

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

See tangent-plane-to-surface.

§150 — Bezout bound for surfaces

If one surface has order $m$ and the other has order $n$, then the projection of the intersection never has order greater than $mn$. (source: appendix6, §150)

Two planes: order-1 projection (a line). Plane + quadric: order-2 projection. Two quadrics: order-4 projection. In general $\le mn$. The 3D analogue of intersection-product-degree-bound. See bezout-bound-surfaces.

§§151–152 — When does the intersection lie in a plane?

The intersection of two surfaces is generically a non-planar curve, but coincidentally planar when both equations satisfy some common linear equation $\alpha z+\beta y+\gamma x=f$. Equivalently: write $z=P(x,y),y=Q(x,y)$ from the two equations, then check whether some linear combination $P+nQ$ leaves only first-degree-in-$z$ and constant terms.

§152’s worked example: right cone $z^2=x^2+y^2$ and elliptic hyperboloid $z^2=x^2+2y^2-2ax-a^2$. Subtracting: $y^2=2ax+a^2$ (a parabola in the $xy$-plane), and $z=x+a$ — so the intersection lies in the plane $z=x+a$. See planar-intersection-condition.

Closing remark — end of the Appendix

These two books have prepared the way for that science. (source: appendix6, §152)

Where “that science” is the calculus, the analysis of the infinite proper. Book II’s surfaces appendix is the final piece of the pre-calculus apparatus: with surfaces, intersections, tangent planes, and projections in hand, the next step is differentiation.

Cross-references

• §150’s $mn$ bound lifts intersection-product-degree-bound from curves to surfaces (modulo projection — the projection bound applies, the actual count is more subtle since spatial intersection can fail to project bijectively).
• §§147–149’s tangent-plane construction is the surface analogue of tangent-by-translation (Book II §§286–291).
• The cone-sphere tangency (§§143–146) is a 3D close cousin of asymptotes-of-hyperbola’s tangent-at-infinity geometry — both encode coincidence of two intersections as double roots of the section eliminant.

Figures

Figures 148–149

Related pages

• non-planar-curves-by-projection
• intersection-of-two-surfaces
• tangency-of-surfaces
• tangent-plane-to-surface
• bezout-bound-surfaces
• planar-intersection-condition
• appendix-4-on-the-change-of-coordinates
• appendix-5-on-second-order-surfaces
• intersection-product-degree-bound
• tangent-by-translation
• oblique-plane-section-method