parabolic curves concentrate all parallel falling rays on a single focal point. Example in nature is path of a projectile. Example in industry is profile of a satellite dish antenna. If a cone is cut parallel to its edge, it is a parabola. Parabola does not have an opposite curve
hyperbolic curves. a straight line can pass through a plane hyperbolic curve if tilted at a particular angle and rotated around a particular axis. cooling tower of a thermal plant, sitting stool made of cane sticks are examples of hyperbola. If a cone is cut parallel to its axis, it a hyperbola.
caternary curve when you hang a string tied at two ends. In inverted form, it is used to make bridges of stones without cement. They just support each other.
an ellipse is obtained by cutting a double cone at an angle. It has two focal points. A ray projected from one focal point to the circumference always reflects to the other focal point.
cycloid, path covered by a reflector on a tyre of a bicycle. or path covered by a point on a circle rotating along a straight line or on another circle. such cycloids are also made by spirographs. if a ball is dropped along a cycloid path, it takes minimum time to reach the other end as compared to a straight path or a circular path. It has one more property. If two balls are dropped along a cycloid from either side from different heights, they meet exactly at the lowest point. That means, no matter from which height you drop a ball along a cycloid, it takes same time to reach the bottom.
Hypocycloid
Hypotrochoid
Epitrochoid
Lissajous figures