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# Basic Mathematics

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### Basic Mathematics

1. 1. Basic mathematics for geometric modeling
2. 2. Coordinate Reference Frames <ul><li>Cartesian Coordinate (2D) </li></ul><ul><li>Polar coordinate </li></ul>x y (x, y) r 
3. 3. Relationship : polar & cartesian Use trigonometric, polar  cartesian x = r cos  , y = r sin  Cartesian  polar r =  x 2 + y 2 ,  = tan -1 (y/x) x P Y x y r   r x y P
4. 4. 3D cartesian coordinates x y z Right-handed 3D coordinate system z x y
5. 5. POINT <ul><li>The simplest of geometric object. </li></ul><ul><li>No length, width or thickness. </li></ul><ul><li>Location in space </li></ul><ul><li>Defined by a set of numbers (coordinates) e.g P = (x, y) or P = (x, y, z) </li></ul><ul><li>Vertex of 2D/ 3D figure </li></ul>
6. 6. <ul><li>distance and direction </li></ul><ul><li>Does not have a fixed location in space </li></ul><ul><li>Sometime called “displacement”. </li></ul>VECTOR
7. 7. VECTOR <ul><li>Can define a vector as the difference between two point positions. </li></ul>x y P Q x1 x2 y1 y2 V V = Q – P = (x2 – x1, y2 – y1) = (Vx, Vy) Also can be expressed as V = Vx i + Vy j Component form
8. 8. VECTOR : magnitude & direction <ul><li>Calculate magnitude using the Pythagoras theorem  distance </li></ul><ul><ul><li>|V| =  Vx 2 + Vy 2 </li></ul></ul><ul><li>Direction </li></ul><ul><ul><li> = tan -1 (Vy/Vx) </li></ul></ul>
9. 9. <ul><li>Example 1 </li></ul><ul><li>If P(3, 6) and Q(6, 10). Write vector V in component form. </li></ul><ul><li>Answer </li></ul><ul><li>V = [6 - 3, 10 – 6] </li></ul><ul><ul><li>= [3, 4] </li></ul></ul>VECTOR : magnitude & direction Q V
10. 10. <ul><li>Example 1 (cont) </li></ul><ul><li>Compute the magnitude and direction of vector V </li></ul><ul><li>Answer </li></ul><ul><li>Magnitud |V| =  3 2 + 4 2 </li></ul><ul><li>=  25 = 5 </li></ul><ul><li>Direction  = tan -1 (4/3) = 53.13 </li></ul>VECTOR : magnitude & direction
11. 11. Unit Vector <ul><li>As any vector whose magnitude is equal to one </li></ul><ul><li>V = V </li></ul><ul><li>|V| </li></ul><ul><li>The unit vector of V in example 1 is </li></ul><ul><ul><li>= [Vx/|V| , Vy/|V|] </li></ul></ul><ul><ul><li>= [3/5, 4/5] </li></ul></ul>
12. 12. VECTOR : 3D <ul><li>Vector Component </li></ul><ul><ul><li>(Vx, Vy, Vz) </li></ul></ul><ul><li>Magnitude </li></ul><ul><ul><li>|V| =  Vx 2 + Vy 2 + Vz 2 </li></ul></ul><ul><li>Direction </li></ul><ul><ul><li> = cos -1 (Vx/|V|),  = cos -1 (Vy/|V|),  =cos -1 (Vz/|V|) </li></ul></ul><ul><li>Unit vector </li></ul><ul><li>V = V = [Vx/|V|, Vy/|V|, Vz/|V|] </li></ul><ul><li>|V| </li></ul>x y z V Vx Vz Vy
13. 13. Scalar Multiplication <ul><li>kV = [kVx, kVy, kVz] </li></ul><ul><li>If k = +ve  V and kV are in the same direction </li></ul><ul><li>If k = -ve  V and kV are in the opposite direction </li></ul><ul><li>Magnitude |kV| = k|V| </li></ul>
14. 14. Scalar Multiplication <ul><li>Base on Example 1 </li></ul><ul><li>If k = 2, find kV and the magnitudes </li></ul><ul><li>Answer </li></ul><ul><li>kV = 2[3, 4] = [6, 8] </li></ul><ul><li>Magnitude |kV|=  6 2 + 8 2 =  100 = 10 </li></ul><ul><li>= k|V| = 2(5) = 10 </li></ul>
15. 15. Vector Addition <ul><li>Sum of two vectors is obtained by adding corresponding components </li></ul><ul><li>U = [Ux, Uy, Uz], V = [Vx, Vy, Vz] </li></ul><ul><li>U + V = [Ux + Vx, Uy + Vy, Uz + Vz] </li></ul>x y V U x y V U U + V
16. 16. Vector Addition <ul><li>Example </li></ul><ul><li>If vector P=[1, 5, 0], vector Q=[4, 2, 0]. Compute P + Q </li></ul><ul><li>answer </li></ul><ul><li>P + Q = [1+4, 5+2, 0+0] = [5, 7, 0] </li></ul>P Q P Q
17. 17. Vector Addition & scalar multiplication properties <ul><li>U + V = V + U </li></ul><ul><li>T + (U + V) = (T + U) + V </li></ul><ul><li>k(lV) = klV </li></ul><ul><li>(k + l)V = kV + lV </li></ul><ul><li>k(U + V) = kU + kV </li></ul>
18. 18. Scalar Product <ul><li>Also referred as dot product or inner product </li></ul><ul><li>Produce a number. </li></ul><ul><li>Multiply corresponding components of the two vectors and add the result. </li></ul><ul><li>If vector U = [Ux, Uy, Uz], vector V = [Vx, Vy, Vz] </li></ul><ul><li>U . V = UxVx + UyVy + UzVz </li></ul>
19. 19. Scalar Product. <ul><li>Example </li></ul><ul><li>If vector P=[1, 5, 0], vector Q=[4, 2, 0]. Compute P . Q </li></ul><ul><li>answer </li></ul><ul><li>P . Q = 1(4) + 5(2) + 0(0) </li></ul><ul><li>= 14 </li></ul>
20. 20. Scalar Product properties <ul><li>U.V = |U||V|cos  </li></ul><ul><li>angle between two vectors </li></ul><ul><ul><li> = cos –1 (U.V) </li></ul></ul><ul><ul><li>|U||V| </li></ul></ul><ul><li>Example </li></ul><ul><li>Find the angle between vector b=(3, 2) and vector c = (-2, 3) </li></ul> U V
21. 21. <ul><li>Solution </li></ul><ul><li>b.c = (3, 2). (-2, 3) </li></ul><ul><li>3(-2) + 2(3) = 0 </li></ul><ul><li>|b| =  3 2 + 2 2 =  13 = 3.61 </li></ul><ul><li>|c| =  (-2) 2 + 3 2 =  13 = 3.61 </li></ul><ul><li> = cos –1 ( 0/(3.61((3.61)) </li></ul><ul><li> = cos –1 ( 0 ) = 90  </li></ul>Scalar Product properties
22. 22. <ul><li>If U is perpendicular to V, U.V = 0 </li></ul><ul><li>U.U = |U| 2 </li></ul><ul><li>U.V = V.U </li></ul><ul><li>U.(V+W) = U.V + U.W </li></ul><ul><li>(kU).V = U.(kV) </li></ul>Scalar Product properties
23. 23. Vector Product <ul><li>Also called the cross product </li></ul><ul><li>Defined only for 3 D vectors </li></ul><ul><li>Produce a vector which is perpendicular to both of the given vectors. </li></ul>x y z a b c c = a x b
24. 24. Vector Product <ul><li>To find the direction of vector C, use righ-hand rules </li></ul>x z A B C x z A B C
25. 25. Vector Product <ul><li>To find the direction of vector C, use righ-hand rules </li></ul>x z A B C A x B x z C A B B x A
26. 26. exercise <ul><li>Find the direction of vector C, (keluar skrin atau kedalam skrin) </li></ul>A B A x B P Q P x Q M N M x N L O L x O
27. 27. <ul><li>If vektor A = [Ax, Ay, Az], vektor B = [Bx, By, Bz] </li></ul><ul><li>A x B = i j k i j </li></ul><ul><li>Ax Ay Az Ax Ay </li></ul><ul><li>Bx By Bz Bx By </li></ul><ul><li>= [ (AyBz-AzBy), (AzBx-AxBz), (AxBy-AyBx)] </li></ul>Vector Product
28. 28. Vector Product <ul><li>Example </li></ul><ul><li>If P=[1, 5, 0], Q=[4, 2, 0]. Compute P x Q </li></ul><ul><li>Solution </li></ul><ul><li>P x Q = i j k i j </li></ul><ul><li>1 5 0 1 5 </li></ul><ul><li>4 2 0 4 2 </li></ul><ul><li>= [ (5.(0)-0.(5)), (0.(4)-1.(0)), (1.(2)-5.(4))] </li></ul><ul><li>= [ 0, 0, -18] </li></ul>P Q
29. 29. Vector Product <ul><li>Properties </li></ul><ul><li>U x V = |U||V|n sin  where n = unit vector perpendicular to both U and V </li></ul><ul><li>U x V = -V x U </li></ul><ul><li>U x (V + W) = U x V+ U x W </li></ul><ul><li>If U is parallel to V, U x V = 0 </li></ul><ul><li>U x U = 0 </li></ul><ul><li>kU x V = U x kV </li></ul>