The conics are the second important class of loci. Unlike lines, conic equations involve the squares of the variables. Conics are a very old topic in mathematics; their names and definitions come from ancient Greece. They are called conics (short for conic sections) since they can be formed by taking slices of a hollow cone at various angles, as show in Figure 1.3.1. In that figure, there are four slices, each steeper than the next. The first horizontal slice gives a perfect circle. The second, slightly tilted slice gives an ellipse. The third slice, precisely at the angle of the cone, give a parabolic. The four slice, steeper than the cone itself, gives a hpyerbolc.
Figure1.3.1.Four Slices of a Cone
Those are the four shapes: circle, ellipse, parabola and hyperbola. I will give two other equivalent definitions for each conic: one instrinsically geometric definition, and one by algebraic equation. For the algebraic equations, I will assume that the conic is centred at the origin. (More general conics can be formed by shifting the conics centered at the origin. Section 1.5 describes these shifts.)
Subsection1.3.2Descriptions of Conics
Figure1.3.2.The Circle
Conic Slice Definition.
A circle is a perfectly horizontal slice of a cone.
Algebraic Definition.
A circle is all points a fixed distrance from the centre. This distance is called the radius. If the circle is centred at the origin and the radius is \(r\text{,}\) then its equation is \(x^2 + y^2 = r^2\text{.}\)
Figure1.3.3.The Ellipse
Conic Slice Definition.
An ellipse is a slice of a cone at an angle greater than zero but less than the angle of the cone.
Algebraic Definition.
An ellipse (centred at the origin) is described by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\text{.}\) For the numbers \(a\) and \(b\text{,}\) which ever is larger is called the semi-major axis which ever is smaller is called the semi-minor axis. These numbers measure the smallest and largest distances from the centre to the edge of the ellipse. These two numbers cannot be equal, since that would produce a circle. In the diagram, I’ve drawn an ellipse with \(a \gt b\) (so that \(a\) is the semi-major axis), but that is not necessary. An ellipse with \(b \gt a\) would be taller than it is wide, and \(b\) would be the semi-major axis.
Figure1.3.4.The Parabola
Conic Slice Definition.
A parabola is a slice of the cone at exactly the angle of the cone.
Algebraic Definition.
A parabola (centred at the origin) is described by the equation \(y = ax^2\text{.}\) The coefficient \(a\) measures the width of the parabola: a larger \(a\) will give a narrower and steeper parabola.
Another Geometric Definition.
From your previous mathematical experience, you may be aware of another way of writing vertically-aligned parabolas. A common way to describe a parabola is by giving a vertex and a leading coefficient. The vertex is the peak of the parabola and the leading coefficient will describe how wide or narrow the parabola is. If the leading coefficient is positive, the parabola will open upwards; if negative, the parabola will open downwards. A parabola with vertex \((a,b)\) and leading coefficient \(c\) is described by the equation \(y = c(x - a) + b\text{.}\) The process of taking a quadratic and writing it in vertex form is called completing the square. This operation should be familiar to you from your high school mathematics, but there is also a reminder on the reference materials.
Figure1.3.5.The Hyperbola
Conic Slice Definition.
A hyperbola is slice of a cone at an angle steeper than the angle of the cone.
Algebraic Definition.
A hyperbola (centred at the origin) is described by the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\text{.}\) The numbers \(a\) amd \(b\) don’t have as immediate a definition as in the ellipse. However, you can notice that the arms of the hyperbola are becoming close to straight lines. The slopes of these lines are \(\frac{a}{b}\) and \(\frac{-a}{b}\text{.}\) These lines (called asymptotes) for the parabola are show int he figure as dotted lines.
One of the major motivating problems for conics and analytic geometry is the problem of celestial motion — how planets, moons, stars, comets and other celestial objects move and orbit around each other. The Greeks assumed, erroneously, that orbits ought to be perfect circles. Johannes Kepler, in the 16th century, correctly observed that orbits take non-circular shapes. He put forward a very convincing theory that orbits have shapes which are conics. This leads to ellipses for objects without escape velocity and hyperbolas for those with escape velocity. Though not perfect (particularly in complicated multi-body problems or when relativistic corrections are included), Kepler’s model is remarkably accurate. Conics are still used as the basic models of orbital paths.
