Prolog

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Ellipse

Fundamental Equation of an 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.

Hyperbola

Fundamental Equation of an 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

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.

Curves