How to find where a parametric curve crosses itself?

Question:

How to find where a parametric curve crosses itself?

Parametric Curves:

Parametric equations describe equations (normally in the form {eq}y = f(x) {/eq}) in which {eq}x {/eq} and {eq}y {/eq} depend on a separate variable. This applies for a number of curves (with respect to {eq}y {/eq}) that cannot be expressed as equation that depends on {eq}x {/eq}.

To find where a parametric curve crosses itself, we first need to understand the expressions defining {eq}x(t) {/eq} and {eq}y(t) {/eq} (assuming that both {eq}x {/eq} and {eq}y {/eq} depend on {eq}t {/eq}. For a parametric curve to cross itself, one condition is that the expression defining {eq}x(t) {/eq} has to have the same value at two different {eq}t {/eq} values (at least). The other part of the condition is that the expression defining {eq}y(t) {/eq} also has to have the same value at two different {eq}t {/eq} values (at least). Therefore, the strategy in this situation is to set up an equation involving the expression defining {eq}x(t) {/eq} on both sides of the equation, with the left side having {eq}t_1 {/eq} replace {eq}t {/eq} and the right side having {eq}t_2 {/eq} replace {eq}t {/eq}. The same thing will have to be done for the expression defining {eq}y(t) {/eq}. If {eq}t_1 {/eq} and {eq}t_2 {/eq} are different from either other, this means that there are two different {eq}t {/eq} values that lead to the same {eq}(x,y) {/eq} coordinate. As a result, these two {eq}t {/eq} values serve as the locations where a parametric crosses itself.