For the Love of Curves!

Oh dear! What curves! The world goes ooh and aah on curves – a female’s curves! But what happens to curves in Mathematics? Eeeu! They are dreadful and monstrous as all things mathematics are made to look to the poor mortals on Mother Earth. But with a little mathematical acumen, we will all realize that a very pertinent question lies when we dig deeper into the concept of a line – the Curvedness of a line.

Reference to free online dictionary.. (poor mathematicians need free mercies!)

A line is a trace of a moving point. When the trace travels in a constant direction it has zero curvature and so it becomes a straight line. When the point travels such that its direction of motion changes, it traces a curved line. Thus we can say that a straight line is a special curve with zero curvature.

Various disciplines within mathematics have given the term different meanings depending on the area of study, so the precise meaning depends on context. However many of these meanings are special instances of the definition which follows. A curve is a topological space which is locally homeomorphic to a line. In everyday language, this means that a curve is a set of points which, near each of its points, looks like a line, up to a deformation.

An arc or segment of a curve is a part of a curve that is bounded by two distinct end points and contains every point on the curve between its end points. When the arc is straight, it is typically called a line segment.

When I teach at the middle level, I teach my students that the shortest distance between two points is a straight line. We do experiments with different lines – straight, curved, perpendicular to check for this result. When we use the term line in our everyday learning of Euclidean Geometry, we simple refer to the straight line. Curved line in Euclidean Geometry has to be specifically mentioned. 
So I hope with this little mathematical background, my readers will.be placed better to appreciate the curves in Mathematics.