Visualization of Conic Sections

Conic sections refer to the geometric figures obtained upon intersecting a cone with a plane making different angles with its axis. In polar coordinates, all conic sections are represented by a single equaion.

Clear Draw

Length of Semi Latus-rectum

Eccentricity

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Brief description

The equation of a conic with a semi latus-rectum length of 'L' and eccentricity of 'e' is given by

\[ r = \frac{l}{1 + e \cos \theta} \]

The corresponding (r, θ) can be converted into cartesian coordinates (x, y) and plotted

Circle

When the plane is perpendicular to the conical axis, the conic section obtained is a circle. The semi latus-rectum is just the radius. Circles have an eccentricity of 0.

Draw
Ellipse

When the plane makes an angle more than the semi-vertical angle of the cone but less than 90^o, the conic section obtained is an ellipse. It like a bloated circle. Ellipses have an absolute eccentricity between 0 and 1 (exclusive).

Draw
Parabola

When the plane makes an angle lesser than the semi-vertical angle of the cone but intersections only a single motif, the conic section obtained is a parabola. They have an absolute eccentricity of 1

Draw
Hyperbola

When the plane intersects both the conical motifs, the conic section obtained is a hyperbola. They have an absolute eccentricity greater than 1.

Draw

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Note:

1. Due to their open-ended nature, parabolas and hyperbolas extend into infinity. They are not efficiently drawn here.
2. Circles and Ellipses of huge lengths can be drawn. However, the curve may seem broken and a lot of time will be required to rescale
3. All draw buttons are disabled during drawing and scaling

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