This week ive been looking at GDI+ in C++. In the lesson i created my first GDI+ window which i drew lines and shapes on (pic). I looked at loading images onto the window which was very easy with help from the MSDN. After having a play with GDI+ i looked at Vector maths. This weeks tasks were maths based and important for later weeks.
Vectors
Magnitude
2D magnitude
Example (Xh,Yh) = (3,4) , (Xt,Yt) = (1,1)
Magnitude is 3-1 (xh-xt) = 2 (Yh-xt) 4-1 = 3
√(2 squared + 3 squared) = 3.606
3D magnitude
Example (1,2,3)
√(1 squared + 2 squared + 3 squared) = 3.741
Scalar
A scalar is doubling or halving the vector
n = (2,4,6) n2 = (4,8,12)
Adding/Subtracting a vector
r s
x|3| x|2| 3+2 = x[5]
y|4| + y|4| = 4+4 = y[8]
z|5| z|1| 5+1 = z[6]
Subtracting 3D vectors
x|3| x|2| 3-2 = x[1]
y|4| – y|6| = 4-6 = y[-2]
z|5| z|1| 5-1 = z[4]
Unit Vectors
A unit vector is a vector that has a magnitude of 1 and is useful when it comes to vector multiplication
Converting a vector into a unit form is called normalizing and is achieved
by dividing a vector’s components by its magnitude
r = [1]
[2]
[3]
||r|| = 1 squared + 2 squared + 3 squared = √14
ru = 1 [1] [0.267] < 1/ √14
√14 [2] = [0.535] < 2/ √14
[3] [0.802] < 3/ √14
The dot product (scalar product)
R = [6] S = [2]
[-3] [4]
[2] [9]
||s||.||r|| cos(β) = 6×2 + (-3)x4 + 2×9 = 18 Dot product
Taking this a step further we can work out the angle between two vectors
||r|| = √62 + (-32) + 22 = 7
||s||= √22 + (-42) + 92 = 10.049
7 x 10.049 x cos(β) = 18
cos(β) = 18
10.049×7 = 0.255
β = cos-1(0.255) = 75.2 degrees
Cross-product (The Vector Product)
The vector product, which is
also called the cross product because of the ‘×’ symbol used in its notation. It
is based on the definition that two vectors r and s can be multiplied together
to produce a third vector t:
r × s = t
a = 8i – 3J + 6k
b = -2i – 4J – 8K
I J K
8 -3 6
-2 -4 -8
I = (-3x-8) – (-4×6) = 24 – - 24 = 48
J = (8x-8) – (-2×6) = -64—12 = -52 Reverse J so answer is 52
K = (8x-4) – (-2x-3) = -32-6 = -38
axb = [48,52,-38]
Filed under: Intro to 3D Graphics | Tagged: 3D, Graphics, Maths, Programming
