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Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 56 / 4 / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The basic parametric surface interrogation enquiries provide the parameter domain and patch adjacency; evaluation of

S (u, v)

; first derivatives

S_{u}, S_{v}

; and, for curvature or error-controlled stepping, second derivatives

S_{uu}, S_{uv}, S_{vv}

. Useful search enquiries also supply conservative bounding volumes over parameter rectangles, subdivision or restriction of a patch, and regularity/singularity information. A local inverse or closest-point enquiry can help seed a search but does not replace global component isolation.

At a regular point,

S_{u}, S_{v}

are independent tangent vectors, and the oriented unit normal vector is

n = \frac{S _{u} \times S _{v}}{∣ S _{u} \times S _{v} ∣} .

(1)

The tangent plane consists of

S + α S_{u} + β S_{v}

. If the cross product is zero, this chart does not define a normal; use a regular alternative chart or treat a true singularity separately.

For two surfaces, collect the four parameters in

w = (u_{1}, v_{1}, u_{2}, v_{2})

and set

F (w) = S_{1} (u_{1}, v_{1}) - S_{2} (u_{2}, v_{2}), J = D F = [A_{1}, - A_{2}], A_{i} = [S_{i, u}, S_{i, v}] .

(2)

At a transverse intersection,

n_{1} \times n_{2} \neq = 0

and

J

has rank three. Hence the implicit function theorem makes

F = 0

locally a curve. Choose its physical unit tangent vector

τ = \frac{n _{1} \times n _{2}}{∣ n _{1} \times n _{2} ∣} .

(3)

With

G_{i} = A_{i}^{T} A_{i}

, the parameter derivatives for physical arc length are

(v _{i}^{'} u _{i}^{'}) = G_{i}^{- 1} A_{i}^{T} τ .

(4)

Since

τ

lies in both tangent planes,

A_{i} (u_{i}^{'}, v_{i}^{'})^{T} = τ

exactly. Thus these two pairs form a vector

d

in the kernel of

J

A practical transversal intersection of two parametric surfaces algorithm is:

Subdivide parameter patches and reject patch pairs whose certified bounding volumes are disjoint. Isolate candidate components, including loops wholly inside a patch; edge crossings or a fixed sampling grid alone are not exhaustive. Obtain corrected seeds for all surviving regular components, using a root search on transverse sections where needed.
At a seed or current point $w$ , evaluate points, derivatives and normals, choose the sign of $τ$ consistently with the preceding step, and predict $w_{p} = w + Δ s d$ and $X_{p} = S_{1} (w) + Δ s τ$ .
Correct all four parameters by solving the four equations

F (w_{c}) = 0, τ \cdot [S_{1} (u_{1, c}, v_{1, c}) - X_{p}] = 0

(5)

with Newton method. The final equation intersects the curve with the plane through

X_{p}

perpendicular to the predicted tangent. Its Jacobian row is

[τ^{T} A_{1}, 0, 0]

; together with

J

it is nonsingular at a sufficiently nearby transverse point. This removes the otherwise free along-curve parameter in the three coincidence equations.

Control $Δ s$ using correction size, conditioning and curvature/chord error bounds. Reject failed steps and halve the step; keep parameters inside their domains or continue through the declared adjacent patch charts. Store the common corrected point and both parameter pairs.
Trace both directions until a surface boundary is reached or a previously traced location on the same branch closes the loop. Maintain component/patch records so later seeds do not duplicate an already traced branch, and continue with every remaining seed.

A finite tolerance supplies a polygonal or fitted-curve approximation; regularity and certified bounds determine its accuracy. Parallel normals, singular charts, branch contacts and coincident patches require separate handling: their intersection may be a singular curve, an isolated point or a two-dimensional region. The transverse marching equations must not be used to claim a unique curve in those cases.

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