The parametric form of a curve is an algebraic representation which expresses the co-ordinates of each point on the curve as a function of an introduced parameter, most frequently . This contrasts with Cartesian form in that parametric equations do not describe an explicit relation between and . This relation must be derived in order to convert from parametric to Cartesian form.
In 3-unit, parametrics focuses on a parametric representation of the quadratic, moving from parametric to Cartesian form and vice-versa, and manipulating geometrical aspects of the quadratic with parametrics. Recognition of other parametric forms is also useful, and more forms are introduced and dealt with in the 4-unit topic, conics.
In specific situations (in the school syllabus, primarily Conic sections), the parametric representation can be useful because:
points on the curve are represented by a single number, not two, simplifying algebra;
some elegant results are possible; for instance, in the standard paramaterisation of the quadratic, the gradient is equal to the parameter, ;
some curves, which cannot be expressed in functional form (for example, the circle, which is neither a function of nor of ) can be conveniently expressed in parametric form;
intuitively, it allows for an easier way to find points on the graph: you can sub in any value for the parameter and instantly find a point, whereas a relational form is not deterministic in the same way.
Furthermore, the parametric form occurs in certain natural phenomena. For instance, using the equations of motion, a thrown ball's location at any time can be calculated using the laws of projectile motion. This is implicitly paramaterising the ball's path by time; to find the shape of the ball's path, (which we know is a parabola) we must use parametrics, to eliminate time, , from the equations.
By inspection, it is obvious that this describes the line . However, what is the formalised approach for doing this?
We are looking for some relation between and which has no in it. In other words, we want to eliminate from the equations. In the above example, we did this by equating the first and second equations, eliminating .
Conceptually, an ellipse is just a 'squished' circle. The parametric form makes this clear:
The cos and sin are still there, but they are now multiplied by different constants so that the and components are stretched differently. We can turn this into the parametric form in a similar fashion to the circle:
which is the standard form of an ellipse, with and intercepts at and , respectively.
3-Unit students are expected to remember the parametric description of a parabola (). They are also expected to know (and/or be able to quickly derive) the equations of the tangent and the normal to the parabola at the point and at the point .
Derivation of equations of the tangent and the normalEdit
Differentiating each parametric equation with respect to ,
Then the gradient can be obtained by dividing by (this is the chain rule):
Note that this result could have been derived without the chain rule, by taking the derivative of the Cartesian form (with respect to ) and solving for . However, the above derivation is faster and more elegant.