python - converting the difference in vectors to between 0 and 1

Question

I'm working with two unit vectors but not sure how to calculate this. I need it so that if they point in the same direction the answer is 1, opposite directions the answer is 0, perpendicular (either up or down) the answer is 0.5, etc.

Examples: For two vectors (1,0) and (-1,0) (so, opposite vectors), the answer I get is 0. For two vectors (1,0) and (1/sqrt(2),1/sqrt(2)) (so, the unit vector pointing at a 45 degree angle) I get 0.25. For two vectors (0,1) and (-1,0) (so, perpendicular vectors) I get 0.5

Thank you for any help with this!

Answer 1

Read about the Dot product In general The dot product of 2 vectors is equal the cosine of the angle between the 2 vectors multiplied by the magnitude (length) of both vectors.

dot( A, B ) == | A | * | B | * cos( angle_A_B ) 

This follows, that the dot product of 2 unit vectors is equal the cosine of the angle between the 2 vectors, because the length of a unit vector is 1.

uA = normalize( A )
uB = normalize( B )
cos( angle_A_B ) == dot( uA, uB )

If 2 normalized vectors point in the same direction, then the dot product is 1, if the point in the opposite direction, the dot product is -1 and if the vectors are perpendicular then the dot product is 0.

In pygame the dot product can be computed by math.Vector2.dot(). If A and B are pygame.math.Vector2 objects:

uA = A.normalize()
uB = B.normalize()
AdotB = uA.dot(uB)

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In the example above, AdotB is in range [-1.0, 1.0]. AdotB * 0.5 + 0.5 is in range [0.0, 1.0] and math.acos(AdotB) / math.pi + 1 maps the angle between A and B linearly to the range [0.0, 1.0].

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Furthermore, pygame.math.Vector2.angle_to() calculates the angle to a given vector in degrees. A value in range [0.0, 2.0] dependent on the angle can be computed by

w = 1 - A.angle_to(B) / 180