Ellipses
by math_explorer, May 26, 2011, 4:28 AM
Since these also seem to kill a lot of weird problems, here's some background!
As you know an ellipse is defined as the locus of all points such that the sum of the distances to two fixed points, the foci, is a constant. An ellipse is a conic section. It has a center, the midpoint of its foci.
The two most important lines/segments associated with an ellipse are its major and minor axes. The major axis is simply the line through the two foci. The minor axis is perpendicular to the major axis, and passes through the center. Their lengths (measured by where they cut the ellipse) are usually denoted
and 
respectively.
A few other numbers:
the distance from one focus to the center is
the eccentricity
is measured as 
, which roughly corresponds to how elongated the ellipse is. When 
we get a circle; when 
we get either a parabola or a line depending on how you approach. Normal ellipses have 
.
Ellipses also have a focus-directrix characterization, a limiting case of which is the parabola. An ellipse can be defined as the locus of points whose ratio of distances to a point and a line is equal to a fixed value. It turns out the ratio is equal to the eccentricity, the point will be one of the foci, and the directrix is parallel to the minor axis.
If
is the distance of the center to the directrix, we have 
.
Okay, now some possibly useful stuff.
PASCAL'S THEOREM works for conic sections too! It follows from just the circle case by a projective transformation (a transformation which preserves collinearity), which can map a circle to an arbitrary conic, but preserves "incidence" (relations like a point being on a line, a line intersecting a curve x times, etc.)
Also a cute lemma: polars wrt ellipses causing harmonic divisions. This lemma would use an affine transformation (you need to preserve the center), but you can generalize even further via projective transformation properties if you don't need the line to pass through the center, because even though projective transformations do not preserve ratios, they preserve cross-ratios and therefore harmonic divisions.
The number of results that can be generalized from our simple property-rich circles is quite amazing.
The center of an ellipse inscribed in a quadrilateral is on the Newton line, because (i) this is true when a quad has an inscribed circle and (ii) Newton lines are preserved by affine transformation since midpoints are preserved and incidence is preserved.
So for example this problem gets solved very neatly yay
If
is on an ellipse, then the angle bisectors of 
are the normal (= orthogonal (= perpendicular)) and tangent lines through . This is why sound waves reflect from one focus to the other in an ellipsoidal room. See http://www.artofproblemsolving.com/blog/48703 for a proof.
Okay this post is long enough
As you know an ellipse is defined as the locus of all points such that the sum of the distances to two fixed points, the foci, is a constant. An ellipse is a conic section. It has a center, the midpoint of its foci.
The two most important lines/segments associated with an ellipse are its major and minor axes. The major axis is simply the line through the two foci. The minor axis is perpendicular to the major axis, and passes through the center. Their lengths (measured by where they cut the ellipse) are usually denoted
and 
respectively.A few other numbers:
the distance from one focus to the center is
the eccentricity
is measured as 
, which roughly corresponds to how elongated the ellipse is. When 
we get a circle; when 
we get either a parabola or a line depending on how you approach. Normal ellipses have 
.Ellipses also have a focus-directrix characterization, a limiting case of which is the parabola. An ellipse can be defined as the locus of points whose ratio of distances to a point and a line is equal to a fixed value. It turns out the ratio is equal to the eccentricity, the point will be one of the foci, and the directrix is parallel to the minor axis.
If
is the distance of the center to the directrix, we have 
.Okay, now some possibly useful stuff.
PASCAL'S THEOREM works for conic sections too! It follows from just the circle case by a projective transformation (a transformation which preserves collinearity), which can map a circle to an arbitrary conic, but preserves "incidence" (relations like a point being on a line, a line intersecting a curve x times, etc.)
Also a cute lemma: polars wrt ellipses causing harmonic divisions. This lemma would use an affine transformation (you need to preserve the center), but you can generalize even further via projective transformation properties if you don't need the line to pass through the center, because even though projective transformations do not preserve ratios, they preserve cross-ratios and therefore harmonic divisions.
The number of results that can be generalized from our simple property-rich circles is quite amazing.
The center of an ellipse inscribed in a quadrilateral is on the Newton line, because (i) this is true when a quad has an inscribed circle and (ii) Newton lines are preserved by affine transformation since midpoints are preserved and incidence is preserved.
So for example this problem gets solved very neatly yay
If
is on an ellipse, then the angle bisectors of 
are the normal (= orthogonal (= perpendicular)) and tangent lines through . This is why sound waves reflect from one focus to the other in an ellipsoidal room. See http://www.artofproblemsolving.com/blog/48703 for a proof.Okay this post is long enough
This post has been edited 3 times. Last edited by math_explorer, Aug 19, 2011, 8:11 AM
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