Wacker Art Intersections of a Cone ## Prolog

### Attention

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## Ellipse

### Parametric Representation of an Ellipse

x = a cos(t); y = b sin(t)

### Examples:

Using the parametric representation of an ellipse, with different values for a and b gives the following graphes.

a = 2.0; b = 1.0;

### Circle

A circle is a special case of an ellipse, with a = b.

a = 1.0; b = 1.0;

a = 1.0; b = 1.5;

## Hyperbola

### Parametric representation of an Hyperbola

Right curve:

x = a cosh(t); y = b sinh(t)

Left curve:

x = -a cosh(t); y = b sinh(t)

Top curve:

x = a sinh(t); y = b cosh(t)

Bottom curve:

x = -a sinh(t); y = b cosh(t)

### Example:

a = 1.0; b = 1.0;

## Parabola

A parabola as function of x.

y=ax2 + bx + c

 a=1; b=2; c=0 a=1; b=0; c=0 a=1; b=-2; c=0 a=-0.5; b=0; c=5 a=4; b=0; c=-2 a=-0.2; b=-2; c=4

### The Vertex of a Parabola

Vertex equation of a parabola:
y=a(x-h)2 + k
y=a(x2-2hx+h2)+k
y=ax2-2ahx+ah2+k

The vertex is the point: (h,k).

### Focus of a Parabola

A parabola with the vertex at the origin (0,0), that is symmetric around the y-axis can be written as:
y = ax2;

The focus point is then defined as f = (0, 1/4a). The vertex is the point (0,0)

#### Example:

with a=1 we get the parabola: y = x2.
The focus is the point (0, 1/4).

Parabola: y=x2; focus = (0, 0.25); vertex = (0.0, 0.0).

### General Construction of a Parabola

A parabola is a set of points that have the same distance between a given point the focus and a given line the base line.

## Curves

The next page is about Curves.

29.Mai 2021 Version 2.0
Copyright: Hermann Wacker Uhlandstraße 10 D-85386 Eching bei Freising Germany Haftungsausschluß