Chapter 6 -- ENTITIES Section ELLIPSE ELLIPSE Command's Parameter Option

The Parameter option of the ELLIPSE command uses the following equation to define an elliptical arc.

p(u)=c+a*cos(u)+b*sin(u)

The variables a, b, c are determined when you select the endpoints for the first axis and the distance for the second axis. a is the negative of 1/2 of the major axis length, b is the negative of 1/2 the minor axis length, and c is the center point (2-D) of the ellipse.

Because this is actually a vector equation and the variable c is actually a point with X and Y values, it really should be written as:

p(u)=(Cx+a*cos(u))*i+(Cy+b*sin(u))*j

where

• Cx is the X value of the point c
• Cy is the Y value of the point c
• a is -(1/2 of the major axis length)
• b is -(1/2 of the minor axis length)
• i and j represent unit vectors in the X and Y directions

In AutoCAD, once the axis endpoints are selected, all you have left to specify is the start and end of the elliptical arc.

When you select the parameter option, you are asked for a start parameter and an end parameter. These values are plugged into the equation to determine the actual start and end points on the ellipse. The rest of the ellipse is filled in between these two points in a counterclockwise direction from the first parameter to the second. The value entered for the parameter u is taken to be degrees for the purposes of obtaining the cos(u) and sin(u).

For example:

Axis endpoint 1 = 0,1

Axis endpoint 2 = 4,1

Other axis distance = 2,0

Start parameter = 270

End parameter = 0

generates the start point at 2,2 and the end point at 0,1 and fills in the ellipse from 2,2 to 0,1 in a counterclockwise direction.