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# Ellipse Approximation: Method 1

### Finding two circular arcs whose normal line at the intersection point is perpendicular to the diagonal connecting the top and side of the pseudo-ellipse

For many practical applications, it is easier to create approximations of ellipses rather than true ellipses. Pairs of circular arcs can be joined at a point of tangency to make smooth pseudo-ellipses. There is some freedom in choosing the radii of the arcs to form a pseudo-ellipse with given major and minor axis lengths.

For example, you can choose the radii such that the arcs are quarter circles. Or, as explained in this article, you can choose the radii such that the normal line *XC₂* and the diagonal *AB* are perpendicular. See diagram below for more detail.

*a*and

*b*are the half-lengths of the major and minor axes respectively. The lengths

*r₂*and

*r₁*are the radii of the larger and smaller circular arcs respectively. The points

*C₂*and

*C₁*are the respective centers of these circles.

*X*is the point where the two circular arcs meet tangently, and the points

*X*,

*C₁*, and

*C₂*are collinear.

By using the right triangle relation leg² + leg² = hypotenuse², you can find the values of

*r₂*and

*r₁*in terms of

*a*and

*b*. This formula is shown in the figure above. The angles of the arcs, φ and θ can be determined once you know

*r₂*and

*r₁*.

**Example:**Suppose you want to make an ellipse approximation that is 8 inches long and 6 inches wide. Thus,

*a*= 4 and

*b*= 3. Using the formula above, you can compute

*r₂*= [16 + 9 + 1*sqrt(25)]/[2*3]

= [25 + 5]/6

= 5

*r₁*= [16 + 9 - 1*sqrt(25)]/[2*4]

= [25 - 5]/8

= 2.5

The angles of these arcs are

φ = arctan(3/4) = 36.87°

θ = arctan(4/3) = 53.13°

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