I have two sets of coordinates:
ax, by, cz
ex, fy, gz
(not using numbers to keep it more generic).
I have a right-handed parametric basis of <N, U, V>. I need to state a vector parametric equation for r(t) in this basis, using the coordinates I have in this plane. I do also have an angle.
I'm really not sure where to go with this one.
To state a vector parametric equation for r(t) in the given right-handed parametric basis <N, U, V>, we can use the coordinates you have (ax, by, cz) and (ex, fy, gz) in this plane, along with the angle you have.
Let's break down the process step by step:
Step 1: Find the direction vector for the given plane
To find the direction vector for the plane, subtract the coordinates of the second point from the coordinates of the first point:
d = (ex - ax, fy - by, gz - cz)
Step 2: Normalize the direction vector
To ensure the direction vector has a length of 1, normalize it by dividing each component by its Euclidean norm:
d_normalized = d / ||d||
Step 3: Find the normal vector to the plane
The normal vector to the plane can be found by taking the cross product of the direction vector with the third basis vector V:
n = d_normalized × V
Step 4: Normalize the normal vector
Similar to step 2, normalize the normal vector to ensure its length is 1:
n_normalized = n / ||n||
Step 5: Find the second basis vector U
The second basis vector U can be found by taking the cross product of the normal vector with the direction vector:
u = n_normalized × d_normalized
Step 6: Define the vector parametric equation
Now that we have the normalized basis vectors N, U, and V, as well as the starting point a(ax, by, cz), we can define the vector parametric equation for r(t) in terms of t:
r(t) = a + N * cos(t * angle) + U * sin(t * angle)
In this equation, a represents the starting point, N represents the normalized normal vector, U represents the normalized second basis vector, and t represents the parameter that determines the position along the curve. The angle indicates the amount of rotation in radians.
By using this equation, you can obtain the vector r(t) at any given t-value within the range of your choice.