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# What is the limiting case of ellipse?

Updated: 12/22/2022

Wiki User

13y ago

The "limiting case" of the ellipse can be a circle, or it can be a straight line.

This isn't a Health question; this is a Math question: specifically, a Geometry question.

Definitions: An ellipse is an 'O' shape, like the slice of an egg.

A "limiting case" is an extreme case, one extreme or the other.

The extreme case of an ellipse, which some people call the "limiting case" of the ellipse, can be a circle, or it can be a straight line. Without going too deeply into the math, when you stretch it out enough, an ellipse tends to become a straight line. When you "round it out", the ellipse tends to look like a circle.

An ellipse is also a conic section: that is, a cut or a slice that you make into a cone. As you make more and more extreme cuts, you get either a circle, or a straight line.

A nice picture of an ellipse, and a good brief explanation, is at Math Warehouse: An ellipse is the locus of all points of the plane, the sum of whose distances to two fixed points add to the same constant. Each fixed point is called a focus. The two fixed points are called the focuses, or foci, the plural of focus.

When the two fixed points of an ellipse are the same point, you get a circle. [Imagine the two points getting closer and closer together, until they become the same point: then, you get a circle. A circle is just a special case of an ellipse.]

If the two focuses get infinitely farther apart, you get a straight line.

Another way of saying this is:

When the eccentricity of an ellipse is zero, the ellipse is a circle. When the eccentricity approaches one, you tend to get a straight line. {Note: if you want to get picky, you never really get to a straight line (that's the infinite case); you just approximate a straight line.}

The equation for an ellipse is PF1 + PF2 = 2a

This is the line segment definition of an ellipse, where PF1 and PF2 are the line segments, which equal the distances from a point P to Focus 1 and Focus 2.

Math Warehouse says the same thing, slightly differently: An ellipse is the set of all points in a plane such that the sum of the distances from T to two fixed points F1 and F2 is a given constant, K. The trigonometric (sine, cosine) and coordinate-geometry (x,y) definitions for an ellipse are at the Math Open Reference siteOther ellipse definitions

There are other ways to define an ellipse. Both use coordinate geometry.

1. Using Trigonometry, with two equations: x = a cos(t)

y = b sin(t)

where t is the parameter and a is the horizontal semi-axis and b the vertical semi-axis

1. Using the formula x2/a2 + y2/b2 = 1

Where a is the horizontal semi-axis and b the vertical semi-axis and the origin is the ellipse center point.

conic sections

The circle is a limiting case of the ellipse, when the slice is made at right angles to the axis, while the parabola is the limiting case of both the ...

http://www.daviddarling.info/encyclopedia/C/conic.html - Cached

Challenge question about ellipse and circles? - Yahoo! UK ...

The limiting case occurs when the radius of curvature of the ellipse is equal to that of the circle. For a smaller radius of curvature, a circle tangent at ...

Degenerate Ellipse - MAA

As long as r is positive, the resulting curve is a legitimate ellipse. In the limiting case of r = 0, the circle is collapsed to a line segment. ...

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