Calculate point, given x, y, angle, and distance?

Question

Excuse my ignorance and use of incorrect terms, but... I have x and y coordinates, and the angle that the entity is facing on a 2D plane. I want to find the correct point, say 5 units in front of the point I have. Examples: If my entity is at 0, 0 and is facing east (0 degrees), my point would be 5, 0. If my entity is at 0, 0 and is facing north (90 degrees), my point would be 0, 5. If my entity is at 0, 0 and is facing north-east (45 degrees), my point would be ???. I can't even figure it out in my head, let alone figure out the formula I need. I assume I need trigonometry, but I'm old and haven't used it since 1997.

Answer 1

If your starting point is (0,0), and your new point is r units away at an angle of θ, you can find the coordinates of that point using the equations x = r cosθ and y = r sinθ. These are the standard equations that let you convert from the rectangular coordinate system (the standard x and y) to the polar coordinate system which works in terms of angles and radius.

For the example you mentioned, we would say r = 5 and θ = 45º. So to calculate our x and y coordinates,

x = r cosθ

x = 5 cos45º

x = 5 (√2 / 2)

x = 5√2 / 2

y = r sinθ

y = 5 sin45º

y = 5 (√2 / 2)

y = 5√2 / 2

Therefore, our point would be ( 5√2 / 2 , 5√2 / 2) or about (3.54, 3.54) if we use the decimal approximation.

Answer 2

At least if I correctly understand your question you are looking for all the points which are 5 units distant from a given point. The locus of such points is, of course, a circle. In the example you chose, the equation of the circle would be x² + y² = 25 and the heading can be figured from the fact that y/x is the tangent of the angle between the point and the x axis. If I misunderstood your question, please respond and clarify.