So I already have x=(1/y)-y^2 and y=t and therefore x=(1-t^2)/t. Then z=(1-2t)/t^2. Is this the parametrization? If so, I have been sitting here for hours trying to find a trig relationship. It just looks too simple.Delta² said:Solve easily for x=f(y). Paremetrization will be y=t, x=f(t).
Wait, so am I thinking about my parametrization wrong? Can I not define x, y, and z in terms of parameter t? In what ways is my answer wrong for my parametrization? I'm sorry if I'm asking so many quick questions but I really would like to understand this concept. So I had solved to x(t), y(t), and z(t) by manipulating the equations of two surfaces, z=x^2-y^2 and z=x^2+xy-1, and from this is gained a parametrization of x(t)=(1-t^2)/t, y(t)=t, and z(t)=(1-2t)/t^2 and I write this as r(t)=<x(t),y(t),z(t)> which is defined when t does not equal 0. Thank you for all your help so far.Delta² said:The parametrization of a curve is done with 1 variable t, but the parametrization of a surface needs 2 variables.
So this form I found is a good final form? I feel like I'm missing something? Thank you for everything! If this is true I can finally sleep before work!Delta² said:Ok i just thought you were trying to parametrize the surface z=... but i see now what you were after.
Parametrization of a curve is the process of representing a curve in terms of one or more parameters, typically denoted by t. This allows for a more systematic and mathematical approach to studying and analyzing curves.
Parametrization of a curve allows for the visualization and manipulation of curves in a more structured and precise manner. It also enables the calculation of important properties such as arc length, curvature, and tangent vectors.
To parametrize a curve, you first need to determine the number of parameters needed. Then, you can express the coordinates of the curve as functions of these parameters. For example, for a 2D curve, you would have two functions for the x and y coordinates, and for a 3D curve, you would have three functions for the x, y, and z coordinates.
A parametric curve is represented by a set of parametric equations, while a cartesian curve is represented by a single equation in terms of x and y. Parametric curves allow for more flexibility and precision, but cartesian curves are often easier to work with and visualize.
Parametrization of a curve has various applications in fields such as engineering, physics, and computer graphics. It is used to model and analyze complex curves and surfaces, as well as to design and control smooth and precise movements in robotics and animation.