Conic Sections Conics, a Family of Similarly Shaped Curves - Properties of Conics Conics, a Family of Similarly Shaped Curves - Properties of Conics
Dandelin's Spheres - proof of conic sections focal properties Proof that conic section curve is the ellipse Proof that conic section curve is the hyperbola Proof that conic section curve is the parabola Conics - a family of similarly shaped curves Conics, a family of similarly shaped curves - properties of conics
By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle c, as is shown in the figure. | | When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse e. | Thus, the circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. | | If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is the parabola p. | | When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone, i.e., if the intersecting plane cuts both cones the hyperbola h is generated. | | |
Dandelin Spheres - proof of conic sections focal properties
Proof that conic section curve is the ellipse In the case when the plane E intersects all generators of the cone, as in down figure, it is possible to inscribe two spheres which will touch the conical surface and the plane. Upper sphere touches the cone surface in a circle k1 and the plane at a point F1. Lower sphere touches the cone surface in a circle k2 and the plane at a point F2.
Arbitrary chosen generating line g intersects the circle k1 at a point M, the circle k2 at a point N and the intersection curve e at a point P. | We see that points, M and F1 are the tangency points of the upper sphere and points, N and F2 the tangency points of the lower sphere of the tangents drawn from the point P exterior to the spheres. | Since the segments of tangents from a point exterior to sphere to the points of contact, are equal | PM = PF1 and PN = PF2. | And since planes of circles k1and k2, are parallel, then are all corresponding generating segments equal | MN = PM + PN is constant. | Thus, the intersection curve is the locus of points in the plane for which sum of distances from the two fixed points F1and F2, is constant, i.e., the curve is the ellipse. | | |
The proof is due to the French/Belgian mathematician Germinal Dandelin (1794 – 1847).
Proof that conic section curve is the hyperbola When the intersecting plane is inclined to the vertical axis at a smaller angle than does the generator of the
cone, the plane cuts both cones creating the hyperbola h which therefore consists of two disjoining branches as shows the right figure. | Inscribed spheres touch the plane on the same side at points F1 and F2 and the cone surface at circles k1and k2. | The generator g intersects the circles k1and k2, at points, M and N, and the intersection curve at the point P. | By rotating the generator g around the vertex V by 360°, the point P will move around and trace both branches of the hyperbola. | While rotating, the generator will coincide with the plane two times and then will have common points with the curve only at infinity. | As the line segments, PF1and PM are the tangents segments drawn from P to the upper sphere, and the segments PF2 and PN are the tangents segments drawn to the lower sphere, then | PM = PF1 and PN = PF2. | Since the planes of circles k1 and k2, are parallel, then are all generating segments from k1to k2 of equal length, so | MN = PM - PN or PF1 - PF2 is constant. | Thus, the intersection curve is the locus of points in the plane for which difference of distances from the two fixed points F1 and F2, is constant, i.e., the curve is the hyperbola. | | |
Proof that conic section curve is the parabola When the cutting plane is parallel to any generator of one of the cones then we can insert only one sphere into the cone which will touch the plane at the point F and the cone surface at the circle k. Arbitrary chosen generating line g intersects the circle k at a point M, and the intersection curve p at a point P. The point P lies on the circle k' which is parallel with the plane K as shows the down figure. By rotating the generator g around the vertex V, the point P will move along the intersection curve.
While the generator approaches position to be parallel to the plane E, the point P will move far away from F. That shows the basic property of the parabola that the line at infinity is a tangent. | The segments, PF and PM belong to tangents drawn from P to the sphere | so, PM = PF. | Since planes of the circles, k and k' are parallel to each other and perpendicular to the section through the cone axis, and as the plane E is parallel to the slanting edge VB, then the intersection d, of planes E and K, is also perpendicular to the section through the cone axis. | Thus, the perpendicular PN from P to the line d, | PN = BA = PM or PF = PN. | | |
Therefore, for any point P on the intersection curve the distance from the fixed point F is the same as it is from the fixed line d, it proves that the intersection curve is the parabola.
Conics - a family of similarly shaped curves A conic is the set of points P in a plane whose distances from a fixed point F (the focus) and a fixed line d (the directrix), are in a constant ratio. This ratio named the eccentricity e determines the shape of the curve. We can see that conics represent a family of similarly shaped curves if we write their equations in vertex form. Recall the method we used to transform equations of the ellipse and the hyperbola from standard to vertex form. We placed the vertex of the curve at the origin translating its graph. Thus, obtained are their vertex equations;
y2 = 2px - (p/a)x2 | - the ellipse and the circle | (for the circle p = a = r) |
| | | y2 = 2px + (p/a)x2 | - the hyperbola | | | Using geometric interpretation of these equations we compare the area of the square y2, formed by the ordinate of a point P(x, y), with the area of the rectangle 2p · x, whose one side is the abscissa x of the point P and the parameter 2p other side, it follows that | | |
- for the ellipse the area of the square is smaller, than the area of the rectangle, - for the parabola is equal, - for the hyperbola the area of the square is greater than the area of the rectangle. The names of curves were given as a result of the above relations, so; - the word “ellipse” (elleipyis) in Greek means “deficiency," - the word “parabola” (parabolh) means “equality” and - the word “hyperbola” (uperbolh) means “excess.” In the given vertex equations we can make following substitutions for;
- the circle | p = a = b = r => e = 0 |
Thus, the equation of conics in vertex form is | y2 = 2px - (1 - e2)x2. | |
The values of e define the curve the conic section makes, such that for e = 0 - a circle, 0 < e < 1 - an ellipse, e = 1 - a parabola, e > 1 - a hyperbola, as shows the above figure.
Conic sections contents Copyright © 2004 - 2020, Nabla Ltd. All rights reserved.