Angle between two lines beginning at one side of the line

Question

I have two lines specified by two points with x and y coordinate each. The first point (where the lines begin) is equal meaning that I have 3 points A, B and C where one line is from A to B and the other one from A to C. I would then want to calculate the angle between the two lines starting at the right side of the first line. The result needn't be accurate I actually only need to know if this angle is greater or less than 180° (π rad). Thanks for any help - you needn't write any code, pseudocode and/or explanation would be great.

Answer 1

Let's define some notation:

A := (a1, a1).
B := (b1, b2).
C := (c1, c2).

Then the determinant D of the matrix

1 a1 a2
1 b1 b2
1 c1 c2

determines whether C lies left or right of the directed line AB [cf. Computational Geometry - Berg, van Kreveld, Overmars, Schwarzkopf - Chapter 1, Exercise 1.4 a)]

Now, you can subtract row 1 from rows 2 and 3:

1 a1     a2
0 b1-a1 b2-a2
0 c1-a1 c2-a2

and calculate D from the first column to get:

(b1-a1)*(c2-a2) > (c1-a1)*(b2-a2)

as the condition representing whether Clies on the left of AC or not. Of course, lying on the right (resp. left) means that the angle is < 180 (resp. > 180)

Example

A = (0, 0)
B = (0, 1)
C = (1, 0)

Then AB is the vertical segment (y axis) and C is clearly on its right side. The condition

(b1-a1)*(c2-a2) > (c1-a1)*(b2-a2)           ineq(1)

becomes:

0 > 1

which is false, meaning that C is not on the left of AB (as expected)

Visualization

To visualize the condition we can translate the three points A, B and C until A becomes (0,0). This is an innocuous transformation because the condition in ineq(1) subtracts a1 and a2 from the coordinates of B and C. With this translation ineq(1) becomes:

b1*c2 > c1*b2                              ineq(2)

Now, lets visualize the three points (ignore for now the brown lines relative to C' = (c'1, c'2)):

The product b1*c2 is the area of the RED rectangle. The product c1*b2 is the area of the GREEN rectangle. The picture shows that RED < GREEN meaning that C is on the right.

Now, mentally move C around, say to C' and visualize the new RED and GREEN rectangles. The inequality remains valid as long as C is on the right.

Not convinced yet? Well, write a program that dynamically reproduces this picture while you change the position of C and voila! (Or study some more Mathematics and solve the book's exercise ;)

Note: Be aware that these rectangles have a sign. In the picture they both have a positive area, however, in general, the products b1*c2 and b2*c1 will have a sign.