Difference between revisions of "Ellipse"

Revision as of 13:56, 13 January 2007

An ellipse is a conic section formed by cutting through a cone at an angle. Equivalently, it is defined as the locus, or set, of all points $P$ such that the sum of the distances from $P$ to two fixed foci (singular focus) is a constant. (The equivalence of these two definitions is a non-trivial fact.)

Ellipses tend to resemble circles which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in Kepler's Laws, the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cut is perpendicular to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.

For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the semimajor axis and the segment connecting the center to a closer end is called the semiminor axis. These two segments are perpendicular. Drawing all four semi-axes divides the ellipse into 4 congruent quarters.

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Using the second definition of an ellipse given above, one may easily construct an ellipse from household materials. To draw an ellipse with two pushpins, a loop of string, pencil, and paper, stick the pushpins in the paper place the string on the paper so that both pushpins are inside it. The pushpins will be the foci of the ellipse, and the length of the string will determine the sum of the distances from a point on the ellipse to the two foci. Hold the pencil on the paper such that the string is taut against the pencil tip and the two pushpins. Then move the pencil tip while keeping the string taut. This will traces out an ellipse.

An ellipse in a Cartesian coordinate system with center $C = (h, k)$ whose axes are parallel to the coordinate axes, with the vertical semi-axis of length $a$ and the horizontal semi-axis of length $b$ is given by the equation $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ . In particular, if the center of the ellipse is the origin this simplifies to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

The three-dimensional counterpart of the ellipse is the ellipsoid.

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 For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat."  The segment connecting the center of the ellipse to one of the "farther away ends" is called the ''[[semimajor axis]]'' and the segment connecting the center to a closer end is called the ''[[semiminor axis]]''.  These two segments are perpendicular.  Drawing all four semi-axes divides the ellipse into 4 [[congruent (geometry)|congruent]] quarters.
-An ellipse may easily be constructed with common household materials, using the second definition of an ellipse. To draw an ellipse with two pushpins, a rubber band, pencil, and paper, stick the pushpins in the paper (these will be the "foci") and wrap the rubber band around the pins. Hold the pencil on the paper such that the rubber band is taut around the pins and pencil tip. Then move the pencil tip while keeping the rubber band taut. This traces out an ellipse, since the distance between the pencil tip and the two foci remains constant (because of the rubber band).
+{{image}}
-The general equation of an ellipse with semi-minor and -major axes a and b and center C(h,k) is <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>, or, when centered at the origin, <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>.
+Using the second definition of an ellipse given above, one may easily construct an ellipse from household materials. To draw an ellipse with two pushpins, a loop of string, pencil, and paper, stick the pushpins in the paper place the string on the paper so that both pushpins are inside it.  The pushpins will be the foci of the ellipse, and the length of the string will determine the sum of the distances from a point on the ellipse to the two foci.  Hold the pencil on the paper such that the string is taut against the pencil tip and the two pushpins. Then move the pencil tip while keeping the string taut. This will traces out an ellipse.
-(definition of eccentricity and polar equation needed)
+An ellipse in a [[Cartesian coordinate system]] with center <math>C = (h, k)</math> whose axes are parallel to the coordinate axes, with the vertical semi-axis of length <math>a</math> and the horizontal semi-axis of length <math>b</math> is given by the equation <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>.  In particular, if the center of the ellipse is the origin this simplifies to <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>.
 The three-dimensional counterpart of the ellipse is the [[ellipsoid]].

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Difference between revisions of "Ellipse"

Revision as of 13:56, 13 January 2007

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