equilateral v. isosceles v. scalene triangles

Triangles can be classified by their relative side lengths.

• An equilateral triangle has three equal sides and three equal angles, all measuring 60º.
• An isosceles triangle has two equal sides and two equal angles.
• A scalene triangle has no equal sides and no equal angles.

Wikipedia on triangles

parabola v. catenary

A blue parabola and a red catenary. Image courtesy of Ask Dr. Math.

Parabola

A parabola is a curve formed by the intersection of a circular cone and a plane parallel to the side of the cone. It can also be defined as the set of points that are the same distance from a line and a point that isn’t on the line. The graph of a quadratic equation of the form  generates a parabola.

Many physical relationships can be formally described by quadratic equations: When air resistance isn’t taken into account, thrown objects follow a parabolic path under the influence of gravity. In addition, parabolic reflectors are used in radio and optical telescopes, microphones, satellite dishes, and car headlights.

Catenary

Stan Wagon riding his square-wheeled trike on an inverted catenary path. Image courtesy of Science News. Catenary curves can look nearly identical to parabolas, but they are defined differently. A catenary is the curve formed by a string suspended from its ends. The general equation for a catenary is

where cosh is the hyperbolic cosine function.

An upside-down catenary is the ideal shape for an arch that only supports its own weight because the arch’s structural elements will experience pure compression, without bending or shear forces. In applications where an arch has to support a large amount of weight, a parabolic curve is preferable. A similar situation applies when the curve opens upwards. Simple suspension footbridges with curved decks are catenaries, while larger suspension bridges with horizontal roadways have parabolic cables.

Ask Dr. Math on parabola v. catenary

rotate v. revolve

Rotation and revolution are types of circular motion that are distinguished by the location of the axis relative to the moving body. When the axis passes through the object, it rotates, but when the axis is outside the object, it revolves. Rotation is synonymous with spinning; revolution is synonymous with orbiting. The Earth rotates about its axis, which runs through the North and South Poles; at the same time, it revolves around the Sun.

Wikipedia on rotation