A parabola is a plane curve whose points are equally distant from a focus and a directrix. In suitable coordinates it has equation , with . Its quadratic geometry can also describe a contour deformation through a saddle point.
A hyperbola is a conic section with two unbounded branches. In suitable coordinates its equation is with . The unit hyperbola is parametrized by , on its right branch, relating it to the hyperbolic functions.
An ellipse is the locus of points whose distances from two fixed foci have constant sum. If the focal distance is and the sum is , its semimajor and semiminor axes are and .
A nondegenerate ellipse has two positive semiaxis lengths. It is a smooth closed curve rather than a point or line segment.
The focal line of an ellipse is the straight line through its two foci.
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A conic section, or simply a conic, is a curve obtained by intersecting a right circular cone with a plane. Depending on the angle and position of the plane relative to the cone, the intersection can generate different types of curves. There are four primary types of conic sections: 1. **Circle**: A circle is formed when the intersecting plane is perpendicular to the axis of the cone. All points on the circle are equidistant from a central point.