Parametric Surfaces and Tangent Planes — Theory
1. Parametric surfaces
A parametric surface is described by a vector-valued function of two variables:
\[
\mathbf r(u,v)
=
\langle X(u,v),Y(u,v),Z(u,v)\rangle.
\]
Each pair \((u,v)\) in the parameter domain gives a point on the surface.
The variables \(u\) and \(v\) act like coordinates on the surface.
2. Surface point
At a selected parameter point \((u_0,v_0)\), the point on the surface is:
\[
\mathbf r(u_0,v_0)
=
\langle
X(u_0,v_0),
Y(u_0,v_0),
Z(u_0,v_0)
\rangle.
\]
This point is the point where the tangent plane is built.
3. Tangent vectors
The two basic tangent vectors are the partial derivatives:
\[
\mathbf r_u
=
\frac{\partial\mathbf r}{\partial u},
\qquad
\mathbf r_v
=
\frac{\partial\mathbf r}{\partial v}.
\]
They point in the directions in which the surface moves when \(u\) or \(v\) changes.
These two vectors span the tangent plane if they are not parallel.
4. Normal vector
A normal vector to the surface is found by taking a cross product:
\[
\mathbf n
=
\mathbf r_u\times\mathbf r_v.
\]
This vector is perpendicular to both \(\mathbf r_u\) and \(\mathbf r_v\).
Reversing the cross product reverses the normal direction:
\[
\mathbf r_v\times\mathbf r_u
=
-(\mathbf r_u\times\mathbf r_v).
\]
5. Regular points
A point on a parametric surface is called regular if the tangent vectors are not parallel.
Equivalently:
\[
\mathbf r_u\times\mathbf r_v\ne \mathbf 0.
\]
If the cross product is zero, the parametrization is singular at that point.
In that case, the tangent plane may not be well-defined.
6. Tangent plane equation
Let
\[
\mathbf r(u_0,v_0)=\langle x_0,y_0,z_0\rangle
\]
and let
\[
\mathbf n=\langle A,B,C\rangle.
\]
Then the tangent plane is:
\[
A(x-x_0)+B(y-y_0)+C(z-z_0)=0.
\]
This plane is the best local flat approximation to the surface at the selected point.
7. Parametric form of the tangent plane
The tangent plane can also be written using the tangent vectors:
\[
\mathbf T(s,t)
=
\mathbf r(u_0,v_0)
+
s\,\mathbf r_u(u_0,v_0)
+
t\,\mathbf r_v(u_0,v_0).
\]
This form is useful for drawing the tangent plane in 3D.
8. First fundamental form
The first fundamental form describes lengths and angles on the surface.
Its coefficients are:
\[
E=\mathbf r_u\cdot\mathbf r_u,
\qquad
F=\mathbf r_u\cdot\mathbf r_v,
\qquad
G=\mathbf r_v\cdot\mathbf r_v.
\]
These values depend only on the tangent vectors.
9. Second fundamental form
The second fundamental form describes how the surface bends relative to the chosen normal.
Let \(\widehat{\mathbf n}\) be the unit normal vector. Then:
\[
L=\mathbf r_{uu}\cdot\widehat{\mathbf n},
\qquad
M=\mathbf r_{uv}\cdot\widehat{\mathbf n},
\qquad
N=\mathbf r_{vv}\cdot\widehat{\mathbf n}.
\]
Reversing the normal changes the signs of \(L\), \(M\), and \(N\).
10. Gaussian curvature
Gaussian curvature is computed from the first and second fundamental forms:
\[
K
=
\frac{LN-M^2}{EG-F^2}.
\]
The sign of \(K\) gives important local information:
- \(K>0\): elliptic behavior, like a sphere near the point.
- \(K<0\): hyperbolic behavior, like a saddle near the point.
- \(K\approx0\): parabolic or nearly flat behavior.
11. Mean curvature
Mean curvature is:
\[
H
=
\frac{EN-2FM+GL}{2(EG-F^2)}.
\]
Mean curvature is related to the average bending of the surface.
If the normal direction is reversed, \(H\) changes sign.
12. Principal curvatures
The principal curvatures \(k_1\) and \(k_2\) can be estimated from \(H\) and \(K\):
\[
k_1,k_2
=
H\pm\sqrt{H^2-K}.
\]
These are the maximum and minimum normal curvatures at the point.
Their product is \(K\), and their average is \(H\):
\[
K=k_1k_2,
\qquad
H=\frac{k_1+k_2}{2}.
\]
13. Example: sphere
A unit sphere can be parametrized by:
\[
\mathbf r(u,v)
=
\langle
\sin v\cos u,\,
\sin v\sin u,\,
\cos v
\rangle.
\]
with
\[
0\le u\le 2\pi,
\qquad
0\le v\le \pi.
\]
Every regular point on the unit sphere has Gaussian curvature:
\[
K=1.
\]
14. Example: torus
A torus can be parametrized by:
\[
\mathbf r(u,v)
=
\langle
(R+a\cos v)\cos u,\,
(R+a\cos v)\sin u,\,
a\sin v
\rangle.
\]
A torus has regions of positive, negative, and zero Gaussian curvature.
This makes it a good example for curvature visualization.
16. Common mistakes
- Using only one tangent vector: a surface tangent plane needs two independent tangent directions.
- Forgetting the cross product: the normal vector comes from \(\mathbf r_u\times\mathbf r_v\).
- Choosing a singular point: if \(\mathbf r_u\times\mathbf r_v=\mathbf 0\), the tangent plane is not well-defined.
- Mixing up parameter values and coordinates: \((u_0,v_0)\) are parameter values, while \((x_0,y_0,z_0)\) is the point in space.
- Reversing orientation without noticing: \(\mathbf r_v\times\mathbf r_u\) reverses the normal direction.
- Expecting mean curvature to be orientation-independent: mean curvature changes sign when the normal is reversed.
- Reading curvature too literally near singularities: curvature formulas require a regular parametrization.
