 |
This is read-only version of the old wiki, feel free to browse it for materials. If you want to share your own materials, please use GeoGebraTube instead. You are also welcome to help us enhance the new wiki. If any questions arise, please contact the webmaster. |
Canonical conics
UNDER CONSTRUCTION
Conic sections are the set of curves that are defined by the intersection of two surfaces in space: the right circular cone and the plane. The study of conics is rich in the connections between formulae, two dimensional constructions (string, ruler, and compass), and three dimensional constructions. What we add here is a geometric interpretation of the standard form of the equation for an ellipse as extended to the four conic sections. We call this standard form the canonical form, because it is the simplest form that brings out all other essential features and because it lends itself to geometric interpretation.
Contents |
Background
All conic sections may be represented by the second degree equation,
formula for second order, first order, cross terms in x and y plus constant
The cross term cxy permits rotation of the curve, while the terms dx and cy permit displacement. Interpreting the constants in geometric terms takes a certain level of abstraction and experimentation. The simpler cannonical form is more useful for illustrating the effect of changing parameters in a formula on how a conic formula appears on paper.
formula for ellipse with offset terms x-x0 and y-y0, and scale constants a and b]
In this formula, x0 shifts the curve left and right, y0 shifts the curve upp and down, a and b change scale in the x and y axis respectively. There is no rotation.
Canonical form of a conic section
This article is concerned with a simple proposal that the formula for an ellipse represents all four conic sections. What is interesting is that the parameters may be interpreted to provide insights into the relationship of the conic sections geometrically and analytically.
| conic section | a | b | x0 | y0 |
| circle | r | r | 0 | 0 |
| ellipse | a | b | 0 | 0 |
| hyperbola | a | b√(-1) | 0 | 0 |
| parabola | a√(h/2) | h | 0 | h |
In the following descriptions, the translational parameters, x¬0 and y0, are set to zero for simplicity, but could have been retained.
Circle
The the circle is parameterized by a single parameter, the radius, r. Parameters a and b, that represent the lengths of the radii in an ellipse, are set equal to r.
Ellipse
The the ellipse is parameterized by parameters a and b, that represent the lengths of the radii in an ellipse. The cononical form given above is normally known as the equation of an ellipse.
Hyperbola
The second parameter, b, of the hyperbola is replaced by an imaginary number. This has the effect of changing the plus sign in the canonical formula to a minus sign, thus converting the formula for an ellipse to the formula for a hyperbola. The geometric interpretation of this is that a hyperbola is an ellipse with one imaginary axis.
Parabola
The parabola is more complex because it requires converting the dependent variable, y, from a quadratic to a linear form. This can be done if one notices that a linear term is available in the expression . The trick is in chosing parameters that eliminate all terms except one term in x2 and one term in y. These parameters are given in the table.
formula substituting table parameters and expanding the squared terms.
Simplifying, this becomes,
formula simplifying by eliminating terms and multiplying through by h.
Taking h to infinity eliminates the y2 term. The result simplifies to the known equation for a parabola.
formula for parabola derived from above
Geometric interpretation
It is interesting to think of a plane intersecting a a pair of right circular cones arranged tip to tip. As the plane rotates, the curve of intersection rotates through four state: circle, ellipse, parabola, and hyperbola. From another prespective, the curve is always an ellipse or hyperbola with four special cases:
- A circle which is produced for an instant as the intersecting plane rotates such that parameters a and b are equal.
- A parabola which is produced for an instant as the intersecting plane rotates such that the parameters a and be go to infinity in a special way.
- A point, which is produced when the intersecting plane passes through the origin and touches neither cone at more than one point.
- Two lines that pass through the origin.
Or to put it in the canonical perspective presented here, the curve is always an ellipse where the hyperbola is a special case of an ellipse having an imaginary axis. This makes one wonder if “imaginary” can have a geometric meaning in the real world.
