Ellipse
Ellipses in 3D geometry is split into two identical half-ellipses. The ellipse is defined by its major and minor axis radius values. Similar to arcs, elliptic segments are defined by a plane in which the ellipse lies, centered at the origin, and parameterized by the angle of rotation. The direction of the ellipse is counterclockwise if the start angle is less than the end angle, otherwise it is clockwise.
Data Format:
center - Center point of the ellipse
major_axis_unit_vect - Direction for the X-axis of the ellipse
norm_axis_unit_vect - Direction for the Y-axis
major_len - The “radius” in the X-direction
minor_leng – The “radius” in the Y-direction
start_ang – The ellipse start angle
end_ang – The end angle for the ellipse
The y-axis can be found as a vector product of norm_axis_unit_vect on the major_axis_unit_vect. In actual examples, the major_len can be less than the minor_len.
Parameterization:
t' (the unnormalized parameter) is
(1 - t) * start_angle + t * end_angle
(x, y, z) = major_len * [cos(t') * vector1] +
minor_len * [sin(t') * vector2] + origin
References:
Faux, I.D., M.J. Pratt. Computational Geometry for Design and Manufacture. Ellis Harwood Publishers, 1983.
Mortenson, M.E. Geometric Modeling. John Wiley & Sons, 1985.