Can anyone tell me how to do this? Just walk me through it:

Find all self-crossings if any for the following curve: x(t) = 1+t^2, y(t) = t^3

Question

Can anyone tell me how to do this? Just walk me through it:

Find all self-crossings if any for the following curve: x(t) = 1+t^2, y(t) = t^3

anonymous · Answer

A point of self crossing means for more than one value of t yields the same (x,y)

anonymous · Answer

Ok, so how should I begin to find that?

anonymous · Answer

Let t1 and t2 be values of t such that the same values of x and y result:
$$x(t _{1})=x(t _{2}) ; y(t _{1})=y(t _{2})$$
Therefore:$$1+t _{1}^{2}=1+t _{2}^{^{2}} ; t _{1}^{^{3}}=t _{2}^{3}$$

anonymous · Answer

So we need to determine if it is possible to have distinct values t1 and t2 which satisfy the above equations. Right?

anonymous · Answer

Got it.

anonymous · Answer

And then how do I proceed?

anonymous · Answer

As can be seen, if $$t _{1}^{3}=t _{2}^{3}$$ then it must be that t1=t2. Right?

anonymous · Answer

We cannot have distinct values of t1and t2.
So to me it seems that this curve does not self intersect.

anonymous · Answer

Do you have the answer? To me it seems this curve does not cross itself and a quick check using graphing tools seems to show the same.

Let me know if you believe there is an error in this as per your information. 
Thanks.

anonymous · Answer

It seems to be so. Thank you.