Find coordinates given original point and slope

[Stephen]

Here's my problem: Given a point (x,y) and a slope m what is the formula to find the point (a,b) which lies d units away from point (x,y) at slope m when point (a,b) is on the left side of (x,y). When (a,b) is on the right side?

I'm not asking someone to just give me the answer, I need to know how to find the answer. This is not for homework, it's for a program i'm writing that requires collision detection and something to position items to a touching position of they happen to collide / intersect. Googling around, I've seen some crazy stuff involving theta and arctangent, which I haven't been over because I'm in pre-calculus.

I also saw the formulas:
x = D/sqrt(1+m^2), and y = mD/sqrt(1+m^2) for right side of (x,y)?
or
x = -D/sqrt(1+m^2), and y = -mD/sqrt(1+m^2) for left side of (x,y)?

however I don't understand how to use these formulas given a point.
If these formulas are the answer could someone please explain to me how to use them?

 

[tkhunny]

Sometimes, one comes to a point where it is time to upgrade one's math skills. Maybe the program is over your head?

1) I suspect you will have difficulty defining "left" and "right". What do you do with a horizontal line?

2) The angle of inclination is given by $\displaystyle \tan(\theta) = \frac{b-y}{a-x} = m$. Therefore, a line containing your point would be $\displaystyle (y-b) = m\cdot (x-a)$

3) All points a given distance, r, from a point (q,s) can be described by $\displaystyle (x-q)^{2} + (y-s)^2 = r^2$

4) Unfortunately this much information narrows the choices down to two possible points. You'll need more to get it to only 1.

5) Actually, I'd examine a rotational mapping. Let us know when you are ready.

 

[Stephen]

In the example above, given the point a, slope of the line, and distance away from point a, there are 2 points, b and c, that satisfy the condition, 5 units away from point a gives a circle of possibilities. Constraining them to a slope (the point must lie on the line of given slope which runs through point a) brings it down to 2 possible points. I have to find out the formula to find the coordinates of point b and c.

I have already accounted for special cases when the slope is undefined or flat. A different process easily corrects collisions on the same axis.

I hope you understand now, there are 2 points, I need to know how to find both of their coordinates.

4) Unfortunately this much information narrows the choices down to two possible points. You'll need more to get it to only 1.

I need to find both points.

Edit:
So I experimented with the formulas and got it working perfectly with this code.

if (zombiesx[x] < zombiesx[y]) {
		//Zombie 1 is left of zombie 2
		zombiesx[y] = zombiesx[y] + ((fixdist)/(sqrt(1+m*m)));
		zombiesy[y] = zombiesy[y] + ((m*fixdist)/(sqrt(1+m*m)));
		zombiesx[x] = zombiesx[x] + (-1*(fixdist/(sqrt(1+m*m))));
		zombiesy[x] = zombiesy[x] + (-1*((m*fixdist)/(sqrt(1+m*m))));
	}
	else if (zombiesx[x] > zombiesx[y]) {
		//Zombie 2 is left of zombie 1
		zombiesx[y] = zombiesx[y] + (-1*(fixdist)/(sqrt(1+m*m)));
		zombiesy[y] = zombiesy[y] + (-1*(m*fixdist)/(sqrt(1+m*m)));
		zombiesx[x] = zombiesx[x] + ((fixdist/(sqrt(1+m*m))));
		zombiesy[x] = zombiesy[x] + (((m*fixdist)/(sqrt(1+m*m))));
	}
	else if (zombiesy[x] > zombiesy[y]) {
		//Zombie 2 is directly above zombie 1
		zombiesx[y] = zombiesx[y] - fixdist;
		zombiesx[x] = zombiesx[x] + fixdist;
	}
	else {
		//Zombie 1 is directly above zombie 2
		zombiesx[y] = zombiesx[y] + fixdist;
		zombiesx[x] = zombiesx[x] - fixdist;
	}

This translates into this mathematically:
Givens:
(X,Y) - starting coordinates
M - slope
D - distance away from (x,y)
Results:
(A,B) - Resulting coordinates that lies distance d from point (x,y) constrained to slope m
If the destination is on the right side:
A = X + D/(sqrt(1+m^2))
B = Y + M*D/(sqrt(1+m*m))
If the destination is on the left side:
A = X + -1*D/(sqrt(1+m*m))
B = Y + -1*M*D/(sqrt(1+m*m))

I believe my math translation is correct, but I'm 100% sure the code snippet works. Tested it with my wii and zombies collide properly and when they intersect, it's immediately corrected.