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AAMAIK

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- TL;DR Summary
- What is the purpose of parametrization

What is a parameter?

To describe the equation of a line, in 2 dimensions, we need a (point on the line + slope to measure slantiness) or two points. Another way: The trajectory of a moving point along the line. Suppose that the moving point initially is at a point of know coordinates r0=(x(t=0), y(t=0), z(t=0)) and that it moves at a constant speed on the line. From the above, the velocity vector has constant components v=(a1,a2,a3) independent of time, and the particle moves along one direction on either side of the initial point. The position of the moving point at any time , t

r=r0+Δr=r0+vΔt=>

x(t)=x(t=0)+a1*t

y(t)=y(t=0)+a2*t

z(t)=z(t=0)+a3*t

If the particle moves at variable speed

x(t)=x(t=0)+ ∫a1(t)dt

y(t)=y(t=0)+ ∫a2(t)dt

z(t)=z(t=0)+ ∫a3(t)dt

With this approach, I can describe the position of any point along the line. But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.

Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.

r=r0+Δr=r0+vΔt=>

x(t)=x(t=0)+a1*t

y(t)=y(t=0)+a2*t

z(t)=z(t=0)+a3*t

If the particle moves at variable speed

x(t)=x(t=0)+ ∫a1(t)dt

y(t)=y(t=0)+ ∫a2(t)dt

z(t)=z(t=0)+ ∫a3(t)dt

With this approach, I can describe the position of any point along the line. But I have problems with this approach, one, if given the x=2 I want to find the corresponding y and z such that the point (x=2, y,z) lies on the line. What if the trajectory of the moving particle does not go through x=2.

Secondly, this approach was based on the fact that the trajectory is along the line (given), it was made up. I don't understand parametrization, I view it as describing things that could be deduced easily from the givens of a moving point not the arbitrary curve in the plane or space it was introduced to serve to describe in the first place.