Parametric representation

**p00ndawg** · April 19th 2010, 03:54 PM

Let p0, p1, and p2 be points in the plane.

a) Specify a parametric representation of a planar curve C(t)
that satisfies C(t0) = p0, C(t1) = p1, and C(t2) = p2.
The components of C should be quadratic polynomials in t.

having alot of trouble with this. Im not quite sure where to begin.

**hollywood** · April 20th 2010, 06:34 AM

If you set the x coordinate to $at^2+bt+c$ , and substitute the x-coordinate part of each of your three conditions, you get three equations of the form $at_i^2+bt_i+c=\text{x coordinate of }p_i$ . Since the $t_i$ and $p_i$ are known values, you have 3 equations in the three unknowns a, b, and c. Then you do the same thing for the y coordinate.

You might find this simpler than solving 3 equations in 3 unknowns (twice):
Lagrange polynomial - Wikipedia, the free encyclopedia

Post again in this thread if you're still having trouble.

- Hollywood

**p00ndawg** · April 20th 2010, 02:37 PM

Originally Posted by hollywood

If you set the x coordinate to $at^2+bt+c$ , and substitute the x-coordinate part of each of your three conditions, you get three equations of the form $at_i^2+bt_i+c=\text{x coordinate of }p_i$ . Since the $t_i$ and $p_i$ are known values, you have 3 equations in the three unknowns a, b, and c. Then you do the same thing for the y coordinate.

You might find this simpler than solving 3 equations in 3 unknowns (twice):
Lagrange polynomial - Wikipedia, the free encyclopedia

Post again in this thread if you're still having trouble.

- Hollywood

yes im having trouble setting this up.
i got help for another question in this thread.

http://www.mathhelpforum.com/math-he...rpolation.html

is it similar?

is it:

x(t) = C0(t)x0 + C1(t)x1 + C2(t)x2
y(t) = C0(t)y0 + C1(t)y1 + C2(t)y2

or am i way off?

**hollywood** · April 20th 2010, 05:11 PM

It is the same as the other thread, except you have both x and y to solve, and you have variables instead of numbers.

Let $p_0=(x_0,y_0)\text{, }p_1=(x_1,y_1)\text{, and }p_2=(x_2,y_2)$ .

Also, let $C(t)=(x(t),y(t))$ .

So we are looking for:

$x(t)\text{ satisfying }x(t_0)=x_0\text{, }x(t_1)=x_1\text{, and }x(t_2)=x_2$ , and

$y(t)\text{ satisfying }y(t_0)=y_0\text{, }y(t_1)=y_1\text{, and }y(t_2)=y_2$ .

Each of these is solved by the same process as the other thread. For the x-coordinate:

$x(t)=x_0\ \frac{t-t_1}{t_0-t_1}\ \frac{t-t_2}{t_0-t_2}+x_1\ \frac{t-t_0}{t_1-t_0}\ \frac{t-t_2}{t_1-t_2}+x_2\ \frac{t-t_0}{t_2-t_0}\ \frac{t-t_1}{t_2-t_1}$

and of course the y-coordinate is similar.

- Hollywood

**p00ndawg** · April 20th 2010, 05:50 PM

Originally Posted by hollywood

It is the same as the other thread, except you have both x and y to solve, and you have variables instead of numbers.

Let $p_0=(x_0,y_0)\text{, }p_1=(x_1,y_1)\text{, and }p_2=(x_2,y_2)$ .

Also, let $C(t)=(x(t),y(t))$ .

So we are looking for:

$x(t)\text{ satisfying }x(t_0)=x_0\text{, }x(t_1)=x_1\text{, and }x(t_2)=x_2$ , and

$y(t)\text{ satisfying }y(t_0)=y_0\text{, }y(t_1)=y_1\text{, and }y(t_2)=y_2$ .

Each of these is solved by the same process as the other thread. For the x-coordinate:

$x(t)=x_0\ \frac{t-t_1}{t_0-t_1}\ \frac{t-t_2}{t_0-t_2}+x_1\ \frac{t-t_0}{t_1-t_0}\ \frac{t-t_2}{t_1-t_2}+x_2\ \frac{t-t_0}{t_2-t_0}\ \frac{t-t_1}{t_2-t_1}$

and of course the y-coordinate is similar.

- Hollywood

oh ok so by doing it for the x and for the y ill get $C(t)=(x(t),y(t))$ ?

where x(t) and y(t) will be a function?

**hollywood** · April 21st 2010, 05:05 AM

Yes, that's correct.

- Hollywood

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