We begin by examining the elementary situation of

x = sin(t)

y = cos (t)

and we allow t to vary from zero to

.

The result of course is the unit circle.

In 3 dimensions allowing

x = cos(t)

y = sin(t)

z =0

you get the perspective of the circle using a homogeneous construction.

Taking things one step further you may allow z =1 . This will produce the isomorphic circle in the

plane 1 unit above the original.

From even an elementary stand point the student should not be spared details of concepts such as

knowing how an extra dimension may play a role in even simple fucntions and their graphs or

merely allow a more analytical perspective.

Keeping in mind the 3 dimension perspective we continue our exploration with the following

situations....

Next lets look at these functions as we complicate the exploration a bit further. If we manipulate the coefficients in front of the sin and cos expressions we add eccentricity to the cirlce curve which is so intrinsic to the trigonometric functions. One notices quite readily that increasing the values in front of the expression for the x parameter creates an ellipse streched in the x direction and intersects the x -axis at the point of the scaled value. An analogous condition occurs for the y parameter.

In the 3-D Plane with z= to some constant, the view appears similar but pushed up into free-space.Using a third dimension should lead to endless possibilites of adventure with parametric and trigonometric functions.

As a closing are a few manipulations are offered

We are reminded of how the function is still a 2 dimensional mapping and more

strongly see the features of it being a disk as well as a complicated curve when viewing the relation

in free-space.

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