General Curves

A curve in 2D or 3D can be descibed by vector function. So, what is vector function? If vector is a set of numbers, then vector fuction is a set of functions. 

In 2D, v(t) = < x(t), y(t) >; in 3D, v(t) = < x(t), y(t), z(t) >. 

For examples, the graph below contains 3 curves:

1. r(t) = < t, t² >
2. r(t) = < cos(t), sin(t) >
3. r(t) = < cos(t), t >

Tick one of the vector functions in the graph, and slide the value of t. Observe the position of the terminal point of the vector v(t), P(x, y). As t changes, point P changes according to the vector. If the trace of point P is connected with a smooth curve, we get the following equations:

No.	Components	Relationship between x and y
1.
2.
3.

In 3D, we don't put the variables in just one equations as in 2D. We leave it as it is in parametric equations. You may slide the value of t below and observe the forming of the curve in 3D (Click here if you wish to have better viewing experience). Even though it is not easy to relate the variables as in 2D,  we may still able to predict the shape of some curves. For examples,  the graph below contains 3 vector functions:

1. r(t) = < 3cos(t), 3sin(t), t >, this gives to a circular curve of radius 3 units. As t increases, z increases as well. As such, point P forms a spring of radius 3 units.
2. r(t) = < cos(t), sin(t), sin(2t) >, since z  is not simply t as in #1, we can only expect a curve that fluctuate between [0, 1] for x, y and z.
3. r(t) = < cos(3t), sin(3t), t >, this looks similar to #1, but with smaller radius (radius = 1 unit) and more intense.

Betty, created with GeoGebra and Thomas Calculus, Pearson.