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Scalars, Points and Vectors (4.2)

Scalars, Points and Vectors (4.2). We will only cover 4.1 - 4.4 The author has managed to make a simple topic be much more complicated than necessary! Scalar - real value Point / Vertex - a location in space (3D) Vector - an entity with direction and magnitude does not have a fixed location

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Scalars, Points and Vectors (4.2)

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  1. Scalars, Points and Vectors (4.2) • We will only cover • 4.1 - 4.4 • The author has managed to make a simple topic be much more complicated than necessary! • Scalar - real value • Point / Vertex - a location in space (3D) • Vector - an entity with direction and magnitude • does not have a fixed location • often represented as directed line segments (a ray)

  2. uisGL (3D object library) • The book explains that OpenGL is not OO • This API helps to hide an internal data structure and some mathematical operations • Refer to Documentation for more information • uisGL Data types • float S; • uisVertex P1, P2(x,y,z); • uisVector V1, V2;

  3. Point / Vertex Operations • Point-Point Subtraction • Vector = Point - Point • uisVertex P1(10, 10, 5), P2(7, 7, 9); • V = P1 - P2; • cout << V; • Point-Vector Addition • P1.set(4, 4, 4); • P2= V + P1; • cout << P2; • Calculating Points on a line • Midpoint = (P1 + P2) / 2 • What about? • P3 = (P1 + P2) / 3 • Parametric Form of a Line • float t = 0.5 • P3 = P1 + V * t • P3 = P1 + (P2 - P1) * t

  4. Vector Operations • Magnitude • the length of a vector is a scalar • sqrt(V*V) • len = V1.length(); • Unit Vector • Vector of length one • v / |v| • V1.normalize(); • Vector-Vector Addition • Vector = Vector + Vector • V3 = V1 + V2

  5. Dot product (4.3) • S = V1 * V2 • S = V1.x*V2.x + V1.y*V2.y + V1.z*V2.z • Vectors are perpendicular if V1 * V2 = 0 • dot product of unit vectors equals cosine of angle • cos (theta) = u * v • Remember • cosine of 0 = 1 • cosine of 90 = 0 • cosine of 180 = -1

  6. Cross product (4.4) • cross product of two vectors results in a third vector perpendicular to both • Calculation • (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx) • Sample code • uisVector V, V1, V2 • V = V1 | V2 • Right handed? • Direction of the perpendicular vector is based on the right handed system • V1 | V2 • V2 | V1 • Group Activity • Given three points • Find a vector perpendicular to the plane • Make it a unit vector

  7. Homogeneous Representations 4.5 • 3D points and vectors are represented in 4D • this helps make some future calculations easier using matrix multiplication • P = (x, y, z, 1) • V = (x, y, z, 0) • Note the calculations • adding two vectors results in w = 0 • adding a vector to a point results in w = 1

  8. Parametric Representations 4.5 • line segment • t = time along the line • P’ = P1 + (P2 - P1) * t • P’ = P1(1-t) + P2(t) • perpendicular bisector • M = midpoint • x(t) = Mx - (By - Ay)t • y(t) = My + (Bx - Ax)t • segment length • sqrt ((Bx - Ax)2 + (By-Ay)2)

  9. A plane • Given three points • P = P1 + (P2-P1)t + (P3-P1)*s • Calculate the centroid of a polygon • P = (P1 + P2 + P3) / 3;

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