Tangency of Surfaces

Summary: §§139–146 of the Appendix on Surfaces. A tangency is nothing but the coincidence of two intersections — read off when the projection eliminant from intersection-of-two-surfaces has a double root or a perfect-square factor. Three types: single tangent point (eliminant has a single doubled root), tangency along a curve (eliminant divisible by a perfect square), or tangency in two points (four complex factors). Worked examples: sphere-plane tangency at $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$ (§§140–141), and cone-sphere tangency along a circle for the right cone with $f=a\sqrt{1+n}$ (§§143–146).

Sources: appendix6, §§139–146.

Last updated: 2026-05-12.

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§139 — Tangency = coincident intersections

If for a projection of this kind, we obtain an equation which has two equal roots. The reason for this is that a tangency is nothing but the coincidence of two intersections. (source: appendix6, §139)

When two surfaces just touch without crossing, two of their normally-distinct intersection points coalesce. Algebraically: the projection eliminant has a double root at the tangency point.

Cases:

• Projection has one (doubled) root: surfaces are tangent at a single point.
• Projection equation divisible by a perfect square: surfaces are tangent along a curve (the square root of the perfect-square factor traces the tangency curve in projection).
• Equation has two equal pairs of factors / four complex factors: surfaces are tangent at two isolated points.

§§140–141 — Sphere-plane tangency

Sphere $z^2+y^2+x^2=a^2$ ∩ plane $\alpha z+\beta y+\gamma x=f$. Substituting $z=(f-\beta y-\gamma x)/\alpha$ and clearing denominators gives the projection

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

Solving for $y$:

$$y=\frac{\beta f-\beta \gamma x\pm \alpha \sqrt{a^2({\alpha }^2+{\beta }^2)-f^2+2\gamma fx-({\alpha }^2+{\gamma }^2)x^2}}{{\alpha }^2+{\beta }^2}.$$

Tangency condition: when $f=a\sqrt{{\alpha }^2+{\beta }^2+{\gamma }^2}$, the discriminant under the square root has a double root at the unique tangent point:

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

This formula can be proved in elementary geometry where the tangent plane of a sphere is treated. (source: appendix6, §141)

Geometrically: the plane is tangent when its signed distance from the origin equals the sphere’s radius, with tangent point along the plane normal.

§142 — General tangency rule

First we use the two equations to eliminate one of the variables. Then we determine whether the resulting equation can be written as the product of two linear factors set equal to zero. (source: appendix6, §142)

Factor structure of the eliminant:

Eliminant structure	Tangency
Two complex linear factors	Tangent at a single point
Two real linear factors equal	Tangent along a line
Two equal nonlinear factors / perfect-square factor	Tangent along a curve
Four complex factors	Tangent at two points

§§143–146 — Cone-sphere tangency

Sphere $z^2+y^2+x^2=a^2$ and cone $(f-z{)}^2=m{x}^2+n{y}^2$ with vertex at distance $f$ along the sphere’s center axis. Eliminate $y$:

$$(f-z{)}^2=n{a}^2-n{z}^2+(m-n)x^2.$$

Right cone ($m=n$): the equation becomes

$$z=\frac{f\pm \sqrt{n(1+n)a^2-n{f}^2}}{1+n}.$$

For tangency, the discriminant must vanish, giving

$$f=a\sqrt{1+n}⟹z=\frac{a}{\sqrt{1+n}}.$$

At this $f$, the surfaces are tangent along a circle — the projection equation onto the $xz$-plane has a double root in $z$ that’s still a perfect-square in $(x,z)$ for each $x$.

Scalene cone ($m\ne n$): more careful analysis (§146). If $m<n$: tangency at two points with $x=0,y=\pm a\sqrt{n}/\sqrt{1+n},z=a/\sqrt{1+n}$. If $m>n$: tangency at two points with $y=0$. The tangent points lie where the cone is most narrow — its short axis matches the sphere’s surface.

Geometric reading

Tangency along a curve = inscribed (or circumscribed) sphere/cylinder geometry. Tangency at isolated points = generic surface kissing. The eliminant’s factor structure encodes all the cases via algebraic genericity vs. degeneracy.

Cross-references

• §139 lifts the osculating-circle coincidence-of-intersection idea (Book II Chapter 14) one dimension up: there, an osculating circle “kisses” a curve where two intersection points coalesce; here, surfaces kiss along curves or at points by the same algebraic mechanism.
• The cone-sphere construction §§143–146 underlies classical Apollonius geometry — inscribed cones are the limiting case of cone-sections.
• 3D analogue of the conic-tangency calculations in biquadratic-by-circle-and-parabola and quadratic-by-line-and-circle — both rely on coincident roots of the elimination polynomial.

Related pages

• appendix-6-on-the-intersection-of-two-surfaces
• intersection-of-two-surfaces
• tangent-plane-to-surface
• osculating-circle
• cone-sections
• sphere-sections
• biquadratic-by-circle-and-parabola