Curves

These are 2D shapes that may make up the cross sectional of a surface.

Ellipses

These are shapes such as an oval or circle.

General Form:

{(\frac{x - h}{A})}^{2} + {(\frac{y - k}{B})}^{2} = 1

$A$ and $B$ are constants.

There are two different types of hyperbolas, ones with a vertical asymptote (#Type I )and ones with a horizontal asymptote (#Type II).

As mentioned before these have a vertical asymptote.

General Form:

As mentioned before these have a horizontal asymptote.

General Form:

$A$ , $B$ and $C$ are constants.

The Cross-sectionals of these shapes are either ellipses or points.

They generally follow the form:

{(\frac{x}{A})}^{2} + {(\frac{y}{B})}^{2} + {(\frac{z}{C})}^{2} = c

They generally follow the form:

z = {(\frac{x}{A})}^{2} + {(\frac{y}{B})}^{2}

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Variables can be swapped the variable that is equal to the other two squared summed will be the direction of the concavity (hollow part of the cup).

They generally follow the form:

z = {(\frac{x}{A})}^{2} - {(\frac{y}{B})}^{2}

Pasted image 20230924174202.png

They generally follow the form:

{(\frac{x}{A})}^{2} + {(\frac{y}{B})}^{2} = {(\frac{z}{C})}^{2} + c

Pasted image 20230924174340.png

They generally follow the form:

{(\frac{x}{A})}^{2} + {(\frac{y}{B})}^{2} = {(\frac{z}{C})}^{2} - c

Pasted image 20230924174531.png

They generally follow the form:

{(\frac{x}{A})}^{2} + {(\frac{y}{B})}^{2} = {(\frac{z}{C})}^{2}

Pasted image 20230924174543.png