location on sphere, given start location, velocity, time

[lcarter]

My task is to write an equation to determine the location of a moving point on a sphere given its starting location and velocity and the amount of time that has passed since the point was at the starting location. The sphere may have any radius but its center is always at the origin (0,0,0). There is no gravity and no friction - velocity is constant.

Known variables:
radius of sphere - r
starting location - init_x, init_y, init_z
velocity - speed_x, speed_y, speed_z (in units/second)
time passed - t (in seconds)

Unknown variables:
ending location - final_x, final_y, final_z

I am doing this as a part of a personal experiment for a computer science class. It has been a few years since I took calculus, trigonometry, or geometry, but I should still be able to understand the answer.

Thank you for any help with this.

 

[mmm4444bot]

3-D Parametric Equation

My task is to write an equation to determine the [ending coordinates] of a moving point ... given its starting [coordinates] and [each component's velocity] and the amount of time that has passed ... [all velocities are] constant.

Known variables:

starting location - init_x, init_y, init_z
velocity - speed_x, speed_y, speed_z
time passed - t

Unknown variables:
ending location - final_x, final_y, final_z

I am doing this as a part of a personal experiment ... few years since I took calculus, trigonometry, or geometry ...

Hello Karter:

A calculus student is familiar with a 3-dimensional rectangular coordinate system and vectors. I assume you remember the former and that you will understand the latter in my description below.

Since the velocity is predetermined in each dimension to get from point A to point B, you can forget about the sphere and its radius; they have nothing to do with your experiment. In other words, the moving point will end up wherever the elapsed time and given velocities take it.

(If the given velocities are in proper ratio to move you from one point on a particular sphere to another point on the same sphere, then that is what will happen.)

Moving from point A to point B within in a three-dimensional coordinate system is a sum of three displacements, one in each of the x-, y-, and z-dimensions.

Each of these three displacements can be visualized as a vector (line segment with positive or negative direction). The length of each vector represents the distance of each displacement. The direction (positive or negative) comes from the sign on the corresponding velocity.

Your exercise provides for determining these three displacements in terms of rate (velocity) and time.

DISTANCE = RATE * TIME

In each dimension, adding the displacement to the starting coordinate yields the ending coordinate.

We write functions to do that. The functions that input starting coordinate and output ending coordinate for each dimension follow.

\(\displaystyle x(t) = x + r_x \cdot t\)

\(\displaystyle y(t) = y + r_y \cdot t\)

\(\displaystyle z(t) = z + r_z \cdot t\)

Replace the symbols in these functions with your given values. (Speed is the absolute value of velocity; I switched terms to preserve the sign.)

\(\displaystyle x = init_x\)

\(\displaystyle r_x = velocity_x\)

\(\displaystyle y = init_y\)

\(\displaystyle r_y = velocity_y\)

\(\displaystyle z = init_z\)

\(\displaystyle r_z = velocity_z\)

EG: Given point A (-25, 7, -15)

velocity_x = 4 units per second

velocity_y = -10 units per second

velocity_z = 9 units per second

x(t) = -25 + 4t

y(t) = 7 - 10t

z(t) = -15 + 9t

Elapsed time is t = 30 seconds

x(30) = 95

y(30) = -293

z(30) = 255

The coordinates of point B are (95, -293, 255)

Please feel free to ask specific questions about any of this.

Also, a Google search on 3-d coordinate systems should yield some decent images.

Cheers,

~ Mark

 

[Deleted member 4993]

Re: location on a sphere

If the particle is constrained to stay on the surface of the sphere - then velocity vector - cannot be constant (except for 0). That constraint will apply translate into centripetal/centrifugal accelerations and appropriate forces.

It is just like on a circle a particle cannot move with costant velocity. Constant speed yes - but not constant velocity (difference between scalar and vector quantity).

 

[mmm4444bot]

Re: location on a sphere

If the particle is constrained to stay on the surface of the sphere - then velocity vector - cannot be constant (except for 0) ...

Yup, yup.

A more interesting experiment (to me) would be to find the required velocities to move from a given point A on a particular sphere to another given point B on the same sphere.

Repeating this experiment for several pairs of given points A and B (eg: from a point on the equator to one of the poles, from one latitude to another, from one longitude to another, etc.) would provide data in which patterns might be observed.

Fun stuff to do on a rainy day ...

~ Mark