How to find parametrization in evaluating of line interval
Thus an appropriate parametrization is very imporant part in solving the problem.
Now, lets consider more general case.
Assume that line segment C joins points two points: strat point P(x1, y1, z1) and end point
Q(x2, y2, z2).
How to find a smooth parametrization (Note that parametrization usually is not unique. Usually
there exist infinitely many oarametrizations for the given curve).
Goal: Find r(t)=(g(t), h(t), k(t)),
here
g(t) – x coordinate
h(t) – y coordinate
k(t) – z coordinate
Assuming that 0≤t≤1 we get
g(0)=x1, g(1)=x2
(1)
h(0)=y1, h(1)=y2
(2)
k(0)=z1, k(1)=z2
(3)
We know the point-point equation of line through two points. Apply this equation to (1). We get
𝑥−𝑥1
=
𝑡−0
𝑥2−𝑥1 1−0

𝑥−𝑥1
=𝑡
𝑥2−𝑥1

(x-x1)=t(x2-x1)
g(t)=x=x1+t(x2-x1)
By similar way we have
(4)
h(t)=y=y1+t(y2-y1)
k(t)=z=z1+t(z2-z1)
(5)
(6)
Relations (4-6) are the seeking parametrization

. Do not forget to write 0≤t≤1.
Use formulue (4-6) for the final exam in task 6. (I solved almost third part of the task)
Good luck,
Fuad bey