Surface

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Surface

Parametric Form

A surface is defined as locus of points \( (x,y,z) \) whose Cartesian coordinates \( x,y,z \)are function of two independent parameter, say \( u \) and \( v \).
It is written as
\( x= f(u,v), y= g(u,v), z= h(u,v) \) (i)
For Example
\( x= u+v, y= u-v, z=u.v \) is a surface
The equation given in (i) is called parametric/freedom/explicit form of the surface.

Plane Surface

 
  



w1=[1, 1, 0],         
w2=[0, 1, 1/2];
p=[1, 1, 1];
view.create('parametricsurface3d', [
            (u, v) => p[[0]]+u*w1[[0]]+v*w2[[0]],
            (u, v) => p[[1]]+u*w1[[1]]+v*w2[[1]],
            (u, v) => p[[2]]+u*w1[[2]]+v*w2[[2]],
            [-4, 4],
            [-4, 4]
        ], {
            strokeColor: 'red',
            stepsU: 10,
            stepsV: 10
        });

Cylinder 

  



view.create('parametricsurface3d', [
            (u, v) => 3*Math.cos(u),
            (u, v) => 3*Math.sin(u),
            (u, v) => v,
            [0, 2*Math.PI],
            [-4, 4]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 10
        });

Cone

  



view.create('parametricsurface3d', [
            (u, v) => v*Math.cos(u),
            (u, v) => v*Math.sin(u),
            (u, v) => -5+v,
            [0, 2*Math.PI],
            [0, 6]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 10
        });

Sphere




view.create('parametricsurface3d', [
            (u, v) => 3 * Math.cos(u) * Math.sin(v),
            (u, v) => 3 * Math.sin(u) * Math.sin(v),
            (u, v) => Math.cos(v),
            [0, 2 * Math.PI],
            [0, Math.PI]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 20
        });

Ellipsoid




view.create('parametricsurface3d', [
            (u, v) => 6 * Math.cos(u) * Math.sin(v),
            (u, v) => 4* Math.sin(u) * Math.sin(v),
            (u, v) => Math.cos(v),
            [0, 2 * Math.PI],
            [0, Math.PI]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 20
        });

Torus




view.create('parametricsurface3d', [
            (u, v) => (4+ Math.cos(v)) * Math.cos(u),
            (u, v) =>  (4+ Math.cos(v)) * Math.sin(u),
            (u, v) => Math.sin(v),
            [0, 2 * Math.PI],
            [0, 2*Math.PI]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 20
        });

Mobis strip




view.create('parametricsurface3d', [
            (u, v) => (4+ v*Math.cos(u/2))*Math.cos(u) ,
            (u, v) => (4+ v*Math.cos(u/2))*Math.sin(u) ,
            (u, v) => v*Math.sin(u/2),
            [0, 2*Math.PI],
  [-1,1]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 5
        });

Enneeper Surface




view.create('parametricsurface3d', [
            (u, v) => u - u**3/3 + u*v**2 ,
            (u, v) => -v - u**2*v + v**3 /3,
            (u, v) => u**2 - v**2,
            [-2,2],
  [-2,2]
        ], {
            strokeColor: 'red',
            stepsU: 20,
            stepsV: 60
        });

Test your surface

 
        Function term in u and v:
        x= 
        y= 
        z=

Vector form of surface

A surface is defined as locus of point \( (x,y,z) \) whose position with respect to origin O is function of two independent parameter \( u \) and \( v \).
It is written as
\(\vec{r} = \overline{OP} \)
or \(\vec{r} = (x,y,z) \)
or\(\vec{r} = (f(u,v), g(u,v), h(u,v)) \)
or\(\vec{r} = \vec{r}(u,v) \) (ii)
For Example
\(\vec{r} = (u+v, u-v, u.v) \) is a surface
The equation given in (ii) is called vector/Gauss form of the surface.
In this vector form of the surface, we denote
First order derivatives
\( \frac{\partial \vec{r}}{\partial u} =\vec{r_1} \)
\( \frac{\partial \vec{r}}{\partial v} =\vec{r_2} \)
Second order derivatives
\( \frac{\partial^2 \vec{r}}{\partial u^2} =\vec{r_{11}} \)
\( \frac{\partial^2 \vec{r}}{\partial v^2} =\vec{r_{22}} \)
\( \frac{\partial^2 \vec{r}}{\partial u \partial u} =\vec{r_{12}} \)
And similarly for the higher derivatives.

Implicit form of surface

A surface is defined as locus of points \( (x,y,z) \) whose Cartesian coordinate satisfy an equation of the form
\( F(x,y,z)=0 \) (iii)
For Example
\( x^2+y^2-z^4=0 \) is a surface
The equation given in (iii) is called old/implicit/constraint form of surface.

Plot your surface

Class of Surface

Let \(S: \vec{r} = \vec{r}(u,v) \) be a surface defined in domain D and \( m \) be a positive integer then surface S is said to be of class- \( m \) if \( \vec{r} \) posses non-vanishing continuous partial derivatives up to order \( m \) at each point in the domain D.

Example 1

If a surface is given by
\(\vec{r} = (u+v, u-v, u.v) \)
Then
First order derivatives are
\( \vec{r_1}=(1, 1, v) \)
\( \vec{r_2}=(1, -1, u) \)
Second order derivatives are
\( \vec{r_{11}}=(0,0,0) \)
\( \vec{r_{22}}=(0,0,0) \)
\( \vec{r_{12}}=(0,0,1) \)
Third order derivatives are
All zero
Hence class of the surface S is 2.

Exercise

Explain the parameter u and v in the following surface
  1. r(u,v)=(1, u ,v)
  2. r(u,v)=(cosu, sinu ,v)
  3. r(u,v)=(v cosu, v sinu ,v)
  4. r(u,v)=( cosu sin v, v sinu sin v ,cosv)

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