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2.5. Curves and Surfaces 43
lies in the surface the vector q
also lies in the surface. Since it was obtained by
varying one argument of p, the vector q
is the partial derivative of p with respect
to u, which we’ll denote p
u
. A similar argument shows that the partial derivative
p
v
gives the tangent to the isoparametric curves for constant u, which is a second
tangent vector to the surface.
The derivative of p, then, gives two tangent vectors at any point on the sur-
face. The normal to the surface may be found by taking the cross product of
these vectors: since both are tangent to the surface, their cross product, which is
perpendicular to both tangents, is normal to the surface. The right-hand rule for
cross products provides a way to decide which side is the front, or outside, of the
surface; we will use the convention that the vector
n = p
u
× p
v
points toward the outside of the surface.
2.5.9 Summary of Curves and Surfaces
Implicit curves in 2D or surfaces in 3D are defined by scalar-valued functions of
two or three variables, f : R
2
→ R or f : R
3
→ R, and the surface consists of all
points where the function is zero:
S = {p |f (p)=0}.
Parametric curves in 2D or 3D are defined by vector-valued functions of one vari-
able, p : D ⊂ R → R
2
or p : D ⊂ R → R
3
, and the curve is swept out as t
varies over all of D:
S = {p(t) | t ∈ D }.
Parametric surfaces in 3D are defined by vector-valued functions of two variables,
p : D ⊂ R
2
→ R
3
, and the surface consists of the images of all points (u, v) in
the domain:
S = {p(t) | (u, v) ∈ D }.
For implicit curves and surfaces, the normal vector is given by the derivative
of f (the gradient), and the tangent vector (for a curve) or vectors (for a surface)
can be derived from the normal by constructing a basis.
For parametric curves and surfaces, the derivative of p gives the tangent vector
(for a curve) or vectors (for a surface), and the normal vector can be derived from
the tangents by constructing a basis.