March 30, 2021

Hyperboloid

Hyperboloid is a quadric surface. It is characterized by not being a cone or a cylinder, having a center of symmetry, and intersecting many planes into hyperbolas.
One example of Hyperboloid is given by an equation
$\vec{r}=(x,y,\frac{7x^2+6xy-y^2}{2})$
This graph is shown in the first figure.
The nature of Hyperboloid is visualized below.

Minimal surface

Minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature.
Plane surface and helicoid are example of minimal surface.

Another example of minimal surface is given by
$\vec{r}=(x,y,\log \cos y-\log \cos x)$
This graph is shown in the figure below.

The helicoid

The helicoid is given by an equation
$\vec{r}=(u cosv,u sinv ,v)$
Here, $u$ is rotational length and $v \in [-\pi, \pi ]$ is height
This graph is shown in the figure below.

Cone

Cone is a surface traced by a straight line being revolving around a fixed vector, about a given point.
The fixed vector is called axis
The straight line is called generator
The point is called vertex
Cone is given by a equation
$\vec{r}=(v\cos u, v\sin u,v)$ where v is height of the cone.
Here, $u \in [0,2 \pi ] , v>0$
The graph of a cone is given by
$\vec{r}=(v\cos u, v\sin u,v)$

Cylinder

Cylinder is a surface traced by a straight line being parallel to a fixed vector.
The fixed vector is called axis
The straight line is called generator
Cylinder is given by a equation
$\vec{r}=(a\cos u, a\sin u,v)$
where a is radius and v is height of the cylinder.
Here,
$u \in [0,2 \pi ] , v>0$
The graph of a cylinder is given by
$\vec{r}=(\cos u, \sin u,v)$

Pseduo-sphere

The pseudosphere is a surface of constant negative Gaussian curvature. It is a surface of revolution generated by a tractrix about its asymptote. The parametric equation of pseudosphere is
$x=\sec hu cos v, y=\sec h u sin v, z=u-\tan h$ u;v∈(0,2π)
The coefficients of first fundamental form are
E=tan h²u, F=0, G=sec h²u
The coefficients of the second fundamental form are
$L=-\sec h u \tan h u, M=0, N=\sec h u \tan h u$