How does tan work exactly?

Math and Physics Programming

Started by littletray26 March 28, 2013 05:51 AM

5 comments, last by Paradigm Shifter 11 years ago

littletray26

284

Author

March 28, 2013 05:51 AM

So I've been looking for a way to get a sprite to rotate and face my cursor, and I came across this:

You wan't a sprite to point towards the mouse pointer?

If that is the case you would probably need a algorithm to calculate it. It would depend on the location of the sprite as much as of the mouse pointer.

Let´s say the rotation 0 is straight up then that should be the case when it has the same X as mouse and Y is bigger.

The math here would be, assuming:

pos is position of sprite
mouse is position of mouse

DeltaY = mouse.Y - pos.Y
DeltaX = mouse.X - pos.X

rotation = tan^-1(DeltaY / DeltaX)

Now I understand that it works, but I'm wondering if one of you fine people can help me understand why it works. I'm not that great at math, but I take every chance I can to improve. I need to understand the concept behind tan, and inverse tan, so I can apply it to any other problems I might face that would require it.

Thanks

Just to clarify, I know what tan is. Tan = opposite / adjacent. I know you feed tan the angle, and it returns o/a but why is o/a needed? What is it used for? and I don't know what inverse tan is at all.

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Aressera

3,144

March 28, 2013 07:24 AM

You're better off not using any trig functions and using transformation matrices instead, try this:


Vector2 v = (mousePosition - spritePosition).normalize();
Vector2 w = { v.y, -v.x };

Matrix2x2 m =
{
v.x, w.x
v.y, w.y
};

This should produce proper rotation if your sprite has a rotation of 0 degrees when facing down the X axis. Swap the columns if 0 degrees is facing down the Y axis. You can then take that matrix and use it for the upper-left 2x2 section of a standard 4x4 identity transformation matrix to get a matrix that can be sent to graphics hardware.

What it's doing is computing the rotated x axis (the vector v), then computing a vector perpendicular to v (the vector w) using the 'perp' product (2D cross product analogue). This forms the basis for a rotated coordinate system which can be directly converted to matrix form.

As for your original question, the inverse tangent is a way of computing one of the acute angles of a right triangle with a hypotenuse slope of (DeltaY / DeltaX), where DeltaX and DeltaY are the lengths of the legs of the right triangle. If you invert the expression, inv_tan(DeltaX / DeltaY) gives you the other acute angle. Basically it converts a slope (Dy/Dx) to an angle relative to the X axis. The regular tangent does the opposite, converts an angle to a slope (rise over run).

turch

590

March 28, 2013 02:05 PM

ok, so tan(angle x) = opp / adj

In the image below, tan(x) = 2/1

tan^-1 is the inverse of tan, so if you have the angle, tan(angle) gives you the ratio, and if you have the ratio, tan^-1(ratio) gives you the angle.

In the image below, tan^-1(2/1) = ~1.107 rads (~63.5 deg)

So you have the setup in the image below. Subtracting the position of the image from the mouse gives you the length of the two sides. You then divide them to get the ratio, and pass that to tan^-1 to get the angle (in radians).

The sin and cos formulas work the same but are used when you have the hypotenuse and one of the shorter sides.

There's also sin^-1 and cos^-1, they are also known as arcsine, arccosine, and arctan (asin/acos/atan).

alvaro

21,604

March 28, 2013 02:22 PM

It should be mentioned that the slope doesn't contain the information as to which side the mouse is on, right or left. So when you take the atan(slope) you'll get the same answer for two opposite sides. Then there is the issue of dividing by zero if the line is vertical. There is a function atan2 to which you pass the two coordinates before dividing them and that one does return the correct angle in all cases.

The paragraph above doesn't mean that I condone using angles for these kinds of things. You should use vectors (the way Aressera described), or complex numbers (which are almost the same thing but make it easier to do rotations).