eScience Lectures Notes : Revisal of some basic Maths

u + v = v + u
u + ( v + w ) = (u + v ) + w
there is 0 such as for any vector v, 0 + v = v
for every vector v, there is another vector w such as v + w = 0 . w is written " -v "

2 ) Scalar Multiplication : associative and distributive

(ab)v = a (bv)
1v = v
(a + b)v = av + bv
a(v + w) =av + aw

Given those two operation, me may define a "linear combination" of a set of vector v1, ... vn : any vector of the form

a1v1 + a2v2 + ... anvn

Example of Vector Space : Rn (the set of all ordered n-tuples of real number) is a vector space

Sub example : R3 :


/ r1                     / 1   / 4   / 5
| r2      ......   ex :  | 3 + | 5 = |12
\ r3                     \ 6   \ 2   \ 8

Slide 3 : Plane Vector Splace

One particular Vector Space : The Plane

We chose an origin, and every point of the plane is matched with the vector that links the point to the chosen origin

We may then easily define...

The addition of two vector (the parallelogram rule)

The multiplication of a vector by a scalar

The vector 0 : the one which is matched with the chosen origin

Slide 4 : Affine Space

Affine Space

A set in which geometric operations make sense, but in which there is no distinguished point

(i.e. no origin such as in a vector space)

A affine space consists of a set, called the points of the affine space, an associated vector space and two operations which connect the affine and the vector space

Given 2 Point P and Q, we can form the difference of P and Q which lies in the vector Space

P - Q = v

Given a point P and a vector u, we can add u to P and get another point in the affine space

P = Q + v

Properties to satisfy

(P + v) + u = P + (v + u)

P + u = P if and only if u = 0

Affine combination of the points P and Q by the real number t : a point such as :

P + t (Q-P) ....... if a + b = 1 ... new notation : aP + bQ <=> P + b (Q-P)

NB : convex combination <=> 0 <= t <= 1

if t1 + t2 + ... tn = 1

t1 P1 + t2 P2 + ... tn Pn <=> P1 + t2 (P2 - P1) + ... tn (Pn - P1)

NB : convex affine combination <=> 0 <= ti <= 1

Slide 5 : Various other curiosities

Various other curiosities

Equation of a Line in an Affine Space defined by two point P and Q and t a real

set of point of the form (1-t) P + t Q

Equation of a Plane in an Affine Space defined by three not collinear point P, Q and R

set of point of the form (1-s) ((1-t) P + t Q) +s R

Vector and affine linear subspace are resp. vector and affine space

V : Non empty subset which is stable through the addition and the multiplication by a real

A : such as the set of the difference between any element of A is a linear vector subspace

Example

Vector subspace of R4


/ x
| y
| z
\ 0

Affine Subspace of R4 : standard affine 3 space


/ x                        / x1 - x2
| y              v1 - v2 = | y1 - y2   belongs to the previous 
| z                        | z1 - z2   vector subspace of R4
\ 1                        \ 0

Slide 6 : Cartesian coordinate system

Cartesian coordinate system 2D

The cartesian plane : set of point labeled with coordinate (x, y) is an affine space

Two usual cartesian reference frames

"Maths" frame

"Computer Graphics" frame

Slide 7 : Cartesian coordinate system 3D

Cartesian coordinate system 3D

The Cartesian space : set of point labelled with coordinate (x, y, z) is an affine space

Two different orientations

3 fingers or thumb or corkscrew rules

Right-handed system

Usual in Maths, Physics...

Java system

Left handed system


   / a1     / a2               / a2-a1
 M | b1   P | b2      u = MP = | b2-b1
   \ c1     \ c2               \ c2-c1

Slide 8 : Dot Product and Distance

Dot Product and Distance

a.k.a. Scalar Product


        / x1   / x2
 u.v =  | y1 . | y2  =  x1.x2 + y1.y2 + z1.z2
        \ z1   \ z2

Distance between M and P :

u = MP

Dist (M,P) = ||u|| = SquareRoot(u.u) = SquareRoot( (a2-a1)2 + (b2-b1)2 + (c2-c1)2 )

v = 0 <=> ||v|| = 0 <=> v.v = 0


   / a1     / a2               / a2-a1
 M | b1   P | b2      u = MP = | b2-b1
   \ c1     \ c2               \ c2-c1

Slide 9 : Dot Product and Angle

Dot Product and Angle

u.v = ||u||.||v||.cos(u,v)

if u and v are long enought... (u,v) = acos(u.v / (||u||.||v||)

Trigonometrical circle

sin(0) = 0

sin(PI/2 rad) = 1

cos(0) = 1

sin(a) = opp/hyp

cos(a) = adj/hyp (Trigo + Thales)

if you know the cosinus, you don't know the sign of the angle !

u.v = 0 <=> (u,v) orthogonal, perpendicular (Maths)

u.v <= epsilon <=> (u,v) orthogonal, perpendicular (Computing)

(or u = 0, or v = 0 ...)

Lenght of the perpendical projection of v on u ??

Lenght = u.v / || v ||

Slide 10 : Cross Product

Cross Product

a.k.a the vector product


                  / x1   / x2     / y1.z2 - y2.z1
 u x v = u ^ v =  | y1 x | y2  =  | x2.z1 - x1.z2
                  \ z1   \ z2     \ x1.y2 - x2.y1

u x v is a vector perpendicular to both u and v , which direction is given by the usual right handed rule, and the lenght, by :

|| u x v || = || u || . || v || . sin (u,v)

now we are able to get the angle...

Slide 11 : Exercise from Last year Exam

Exercise from Last year Exam

Question 16 ( 8 marks ) :

Write an algorithm (Java-like pseudocode) that will calculate the point of intersection of a ray (semi segment) and a triangle. The method should return the status of the intersection (intersect or not, and, if not, why not ). If it intersects, then the method should also return the coordinates of the intersection. Use comments in your code to explain the algorithm.

The ray is defined by the origin point O and the direction vector V.
The triangle is defined by the 3 vertices P0, P1, P2.

Tip : if vertices A, B, C and D are on the same plane and if D = A + sAB + tAC then D is inside triangle ABC if and only if s>=0, t>=0 and s+t <= 1

First compute the intersection point M between the ray and the plane of the triangle
Then compute s and t such as M = P0 + s P0P1 + t P0P2
Considering that M is inside the triangle if and only if s >= 0, t >= 0 and s+t <= 1 draw your conclusion

Slide 12 : Exercise from Last year Exam

Exercise from Last year Exam

Question 16 ( 8 marks ) :

Write an algorithm (Java-like pseudocode) that will calculate the point of intersection of a ray (semi segment) and a triangle. The method should return the status of the intersection (intersect or not, and, if not, why not ). If it intersects, then the method should also return the coordinates of the intersection. Use comments in your code to explain the algorithm.

The ray is defined by the origin point O and the direction vector V.
The triangle is defined by the 3 vertices P0, P1, P2.

Tip : if vertices A, B, C and D are on the same plane and if D = A + sAB + tAC then D is inside triangle ABC if and only if s>=0, t>=0 and s+t <= 1

First compute the intersection point M between the ray and the plane of the triangle
Then compute s and t such as M = P0 + s P0P1 + t P0P2
Considering that M is inside the triangle if and only if s >= 0, t >= 0 and s+t <= 1 draw your conclusion

eScience Lectures Notes : Revisal of some basic Maths

Mathematics for Computer Graphics

Cartesian coordinate system 2D

Cartesian coordinate system 3D

Ref : Computer Graphics / Foley - VanDam - Feiner - Hughes

Vector Space

Set of elements called vector, with 2 operations with certain properties:

1 ) Addition of vectors : commutative, associative, with an identity element and each element must have inverses

2 ) Scalar Multiplication : associative and distributive

Given those two operation, me may define a "linear combination" of a set of vector v1, ... vn : any vector of the form

Example of Vector Space : Rn (the set of all ordered n-tuples of real number) is a vector space

Sub example : R3 :

One particular Vector Space : The Plane

We chose an origin, and every point of the plane is matched with the vector that links the point to the chosen origin

We may then easily define...

The addition of two vector (the parallelogram rule)

The multiplication of a vector by a scalar

The vector 0 : the one which is matched with the chosen origin

Affine Space

A set in which geometric operations make sense, but in which there is no distinguished point

(i.e. no origin such as in a vector space)

A affine space consists of a set, called the points of the affine space, an associated vector space and two operations which connect the affine and the vector space

Given 2 Point P and Q, we can form the difference of P and Q which lies in the vector Space

Given a point P and a vector u, we can add u to P and get another point in the affine space

Properties to satisfy

Various other curiosities

Equation of a Line in an Affine Space defined by two point P and Q and t a real

set of point of the form (1-t) P + t Q

Equation of a Plane in an Affine Space defined by three not collinear point P, Q and R

set of point of the form (1-s) ((1-t) P + t Q) +s R

Vector and affine linear subspace are resp. vector and affine space

Example

Vector subspace of R4

Affine Subspace of R4 : standard affine 3 space

Cartesian coordinate system 2D

The cartesian plane : set of point labeled with coordinate (x, y) is an affine space

Two usual cartesian reference frames

"Maths" frame

"Computer Graphics" frame

Cartesian coordinate system 3D

The Cartesian space : set of point labelled with coordinate (x, y, z) is an affine space

Two different orientations

3 fingers or thumb or corkscrew rules

Right-handed system

Usual in Maths, Physics...

Java system

Left handed system

Dot Product and Distance

a.k.a. Scalar Product

Distance between M and P :

Dist (M,P) = ||u|| = SquareRoot(u.u) = SquareRoot( (a2-a1)2 + (b2-b1)2 + (c2-c1)2 )

v = 0 <=> ||v|| = 0 <=> v.v = 0

Dot Product and Angle

u.v = ||u||.||v||.cos(u,v)

if u and v are long enought... (u,v) = acos(u.v / (||u||.||v||)

Trigonometrical circle

sin(0) = 0

sin(PI/2 rad) = 1

cos(0) = 1

sin(a) = opp/hyp

cos(a) = adj/hyp (Trigo + Thales)

if you know the cosinus, you don't know the sign of the angle !

u.v = 0 <=> (u,v) orthogonal, perpendicular (Maths)

u.v <= epsilon <=> (u,v) orthogonal, perpendicular (Computing)

(or u = 0, or v = 0 ...)

Lenght of the perpendical projection of v on u ??

Lenght = u.v / || v ||

Cross Product

a.k.a the vector product

u x v is a vector perpendicular to both u and v , which direction is given by the usual right handed rule, and the lenght, by :

|| u x v || = || u || . || v || . sin (u,v)

now we are able to get the angle...

Exercise from Last year Exam

Question 16 ( 8 marks ) :

The ray is defined by the origin point O and the direction vector V. The triangle is defined by the 3 vertices P0, P1, P2.

Tip : if vertices A, B, C and D are on the same plane and if D = A + sAB + tAC then D is inside triangle ABC if and only if s>=0, t>=0 and s+t <= 1

First compute the intersection point M between the ray and the plane of the triangle

Then compute s and t such as M = P0 + s P0P1 + t P0P2

Considering that M is inside the triangle if and only if s >= 0, t >= 0 and s+t <= 1 draw your conclusion

Exercise from Last year Exam

The ray is defined by the origin point O and the direction vector V.
The triangle is defined by the 3 vertices P0, P1, P2.

The ray is defined by the origin point O and the direction vector V.
The triangle is defined by the 3 vertices P0, P1, P2.