Difficulty: Intermediate | Prerequisites: Familiarity with quadric surfaces, basic vector differentiation, and the dot product (Chapter 12 material in Stewart).
When two surfaces meet in 3D space, they typically share a curve. Describing that curve as a vector function is one of the core skills in Calc III, because it lets you apply everything you know about vector calculus (arc length, curvature, line integrals) to a real geometric object. Separately, when two parametric curves cross at a point, the angle between their tangent vectors at that point tells you how sharply they intersect. Both ideas show up on quizzes and exams regularly, and they tie together surface geometry with the vector-function toolkit.
To find the curve where two surfaces intersect, parameterise one variable (usually with trig for an ellipsoid) and substitute into the other equation. To find the angle between two curves at a shared point, compute each curve's tangent vector at that point and use the dot-product formula for the angle between vectors.
Ellipsoid
A quadric surface of the form x²/a² + y²/b² + z²/c² = 1. All cross-sections are ellipses. When a = b = c it is a sphere.
Think of it as a stretched or squashed sphere.
Parabolic cylinder
A surface of the form y = x² (or any single variable equals a quadratic in another). It extends infinitely in the direction of the missing variable, so it looks like a parabola extruded along an axis.
In simple terms, take a parabola and slide it along a straight line to sweep out a surface.
Curve of intersection
The set of points that lie on both surfaces simultaneously. Described as a vector function r(t) by choosing a parameter and expressing each coordinate in terms of it.
Helix
A space curve that spirals around an axis at constant radius. Standard form: r(t) = a cos(t) i + a sin(t) j + bt k (or with sin and cos swapped between components).
Think of it as wrapping a line around a cylinder, like threads on a screw.
Angle of intersection (of two curves)
The angle θ between the tangent vectors of two curves at their point of intersection. Computed using cos(θ) = |r₁'(t₁) · r₂'(t₂)| / (|r₁'(t₁)| |r₂'(t₂)|). Some textbooks omit the absolute value, giving the angle between 0 and π; others use it so the angle is between 0 and π/2.
The general strategy: choose a parameter for one or two of the coordinates, substitute into both surface equations, and solve for the remaining coordinate(s).
Worked Example (Quiz Q3)
Surfaces: the top half of the ellipsoid x² + 6y² + 6z² = 36, and the parabolic cylinder y = x².
Step 1: Let x = t. Then y = t² (from the cylinder equation).
Step 2: Substitute into the ellipsoid equation to find z.
t² + 6(t²)² + 6z² = 36
t² + 6t⁴ + 6z² = 36
6z² = 36 − t² − 6t⁴
z² = (36 − t² − 6t⁴)/6
Since we want the top half, z ≥ 0:
z = √((36 − t² − 6t⁴)/6)
Step 3: Write the vector function.
r(t) = t i + t² j + √((36 − t² − 6t⁴)/6) k
The domain of t is restricted to values where 36 − t² − 6t⁴ ≥ 0.
Strategy notes
When one surface is a cylinder (missing a variable), it directly gives one coordinate in terms of another. Use that substitution first.
For ellipsoids and spheres, trigonometric parameterisation (x = a cos t, y = b sin t) is sometimes cleaner, but not always necessary.
"Top half" means z ≥ 0, so take the positive square root. "Bottom half" means z ≤ 0.
When two parametric curves meet at a point, the angle between them is defined as the angle between their tangent vectors at that point.
Worked Example (Quiz Q4)
The helix r₁(t) = 4 cos(t) i + sin(t) j + t k intersects the curve r₂(t) = (4 + t) i + 6t² j + 5t³ k at the point (4, 0, 0).
Step 1: Find the parameter values.
For r₁: 4 cos(t) = 4 gives cos(t) = 1, so t = 0. Check: sin(0) = 0 and t = 0. Point is (4, 0, 0). Confirmed.
For r₂: 4 + t = 4 gives t = 0. Check: 6(0)² = 0 and 5(0)³ = 0. Point is (4, 0, 0). Confirmed.
Step 2: Compute the tangent vectors.
r₁'(t) = −4 sin(t) i + cos(t) j + k. At t = 0: r₁'(0) = 0 i + 1 j + 1 k = ⟨0, 1, 1⟩.
r₂'(t) = 1 i + 12t j + 15t² k. At t = 0: r₂'(0) = 1 i + 0 j + 0 k = ⟨1, 0, 0⟩.
Step 3: Use the dot-product angle formula.
cos(θ) = (r₁' · r₂') / (|r₁'| |r₂'|)
r₁' · r₂' = (0)(1) + (1)(0) + (1)(0) = 0.
Since the dot product is 0, the tangent vectors are perpendicular.
θ = π/2 (90°).
Key observations
The two curves may use different parameter values at the intersection point. Always solve for each parameter separately.
A dot product of zero means the curves cross at right angles. This is a clean result worth checking, not a sign of error.
If the problem asked for the angle between the curves (not the tangent lines), the answer is the same, because the angle between curves is defined via their tangent vectors.
Dot product angle formula
cos(θ) = (a · b) / (|a| |b|)
where a · b = a₁b₁ + a₂b₂ + a₃b₃ and |a| = √(a₁² + a₂² + a₃²).
Ellipsoid (standard form)
x²/a² + y²/b² + z²/c² = 1
Parabolic cylinder
y = x² (extends along the z-axis)
Helix (standard form)
r(t) = a cos(t) i + a sin(t) j + bt k
Radius a, pitch 2πb.
Students sometimes confuse intersection of curves with collision of curves. Two curves intersect if they pass through the same point, possibly at different parameter values. They collide only if they are at the same point at the same parameter value.
When parameterising the curve of intersection, students often forget to check that the parameterisation satisfies both surface equations. Always substitute back.
A common mistake is computing the angle between position vectors rather than tangent vectors. The angle of intersection uses derivatives, not the vectors r₁ and r₂ themselves.
Students occasionally forget to restrict z ≥ 0 (or z ≤ 0) when the problem specifies the top or bottom half of a surface.
⚠️ Intersection-of-surfaces problems test whether you can move between implicit surface equations and parametric vector functions. This skill recurs throughout the course.
⚠️ Angle-of-intersection questions are quick to grade and common on quizzes. The dot-product formula must be automatic.
⚠️ Watch for problems that specify "top half" or "bottom half" of a surface. Missing this restriction loses easy marks.
True or False: To find the curve of intersection of two surfaces, you need to eliminate the parameter. (False: you introduce a parameter and express coordinates in terms of it.)
Fill in the blank: The angle between two curves at their intersection is the angle between their ______ at that point. (tangent vectors)
True or False: If two curves have perpendicular tangent vectors at their intersection, the dot product of those tangent vectors is 1. (False: it is 0.)
Fill in the blank: A parabolic cylinder y = x² extends infinitely along the ______ axis. (z)
Q: Find a vector function for the curve of intersection of the sphere x² + y² + z² = 4 and the plane z = 1.
A: With z = 1, the sphere gives x² + y² = 3. Parameterise: x = √3 cos(t), y = √3 sin(t), z = 1. So r(t) = √3 cos(t) i + √3 sin(t) j + 1 k, for t in [0, 2π).
Q: The curves r₁(t) = t i + t² j + t³ k and r₂(t) = sin(t) i + sin(2t) j + t k both pass through the origin. Find the angle of intersection.
A: At the origin, both curves have t = 0. r₁'(t) = ⟨1, 2t, 3t²⟩, so r₁'(0) = ⟨1, 0, 0⟩. r₂'(t) = ⟨cos(t), 2cos(2t), 1⟩, so r₂'(0) = ⟨1, 2, 1⟩. Dot product: 1. Magnitudes: 1 and √6. cos(θ) = 1/√6, so θ = arccos(1/√6) ≈ 65.9°.
Q: Find a vector function for the curve where the paraboloid z = x² + y² meets the plane z = 9.
A: Setting z = 9: x² + y² = 9. Parameterise: x = 3 cos(t), y = 3 sin(t), z = 9. So r(t) = 3 cos(t) i + 3 sin(t) j + 9 k.
Curves of intersection lead directly into line integrals and Green's/Stokes' theorems later in the course, where you integrate a function along a curve that sits on a surface.
The angle-of-intersection concept generalises to the angle between a curve and a surface (using the tangent vector and the surface normal), which appears in later chapters on tangent planes.
Parameterising intersections also prepares you for finding boundaries of surface integrals, where you need the boundary curve in vector form.
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