For questions about circles, ellipses, hyperbolas, and parabolas. These curves are the result of intersecting a cone with a plane.
A conic section is a smooth planar curve that is the result of intersecting a cone with a plane. There are commonly four conic sections: circles, ellipses, parabolas, and hyperbolas.
We can construct these conic sections analytically. The solutions to the equation $x^2+y^2=z^2$ give us a cone in three-dimensional space. An plane in three-dimensional space that goes through a point $p=(x_0,y_0,z_0)$ and has normal vector $\langle a,b,c \rangle$ is given by the equation
$$a(x-x_0)+b(y-y_0)+c(z-z_0)=0\,.$$
Finding the common solutions to the equation of the cone and equation of the plane for various choices of $p$ and normal vector will—after a change in coordinates to write them as planar curves—lead you to the (potentially) familiar equations
- Circle: $x^2+y^2 = r^2$
- Ellipse: $ax^2 + b y^2 = r^2$, where $a,b>0$
- Parabola: $ax^2 +by = r^2$, where $a\neq 0$
- Hyperbola: $ax^2 - by^2 = r^2$, where $a,b>0$
There are also geometric constructions of the conic sections. For example, a circle is the set of all points that are a fixed distance from a given points. An ellipse is the set of all points .... The construction of Dandelin spheres (see Wikipedia) unifies the analytic and geometric constructions of conic sections.
