Useful Curves

Where do these 2 curves intersect?

If I am given 2 curves:
C1(t) = <2t, (t^2)+t, t^3> and
C2(t) =
How do I find their intersection point?

We have 2 curves given as functions of a parameter. To find where they intersect, we clearly need the x component of C1 = the x of C2, and similarly for the y and z component. The key point is that this may not happen at the same t value for each curve. In other words, the “t” in C1 is not the same “t” as in C2.
To give a quick example of why that is so, consider these 2 parametric representations of the unit circle:
1. x = cos t, y = sin t and
2. x = sin t, y = cos t.
These curves both go through the point x = 1, y = 0: that happens for the first one when t = 0 , but it happens at t = pi/2 for the second.

Therefore, we need to use a different letter for the parameter in each and then solve..
So let C1 = <2t, (t^2)+t, t^3> and
let C2 =
Now equate components:
The x tells us we need 2t = s. The y says we need t^2+t = s^2-3s+4.
Substitute s = 2t into the second and we get a quadratic in t:
-3t^2+7t-4=0. Solving gives t = 1 and 4/3. Since s=2t we get that the x and y components are equal when t=1 and s=2, and also when t=4/3 and s= 8/3.
But we haven’t yet used the requirement for the z components to be equal. We need: t^3 = s^2+s-5. Testing the two answers that work for x=y, we find that only t=1, s= 2 makes the z components work as well. Therefore, the answer is that the curves both pass through the point given by t=1, s=2, and that corresponds to x = 2 y = 2, z =1.

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About the Author:

Choose DAY Elevator & Lift to buy curved stair lifts for residential use. We supply residential stair lifts and various other residential accessibility solutions for our customers in the Tri State Area.

Article Source: ArticlesBase.com – Choosing Curved Stair Lifts For Residential Use