2D Computer Graphics Diego Nehab Summer 2020 IMPA 1 Inflection - PowerPoint PPT Presentation

2D Computer Graphics Diego Nehab Summer 2020 IMPA 1

Inflection points and double points

The coordinates P G of P in G are T P F F P F G P G F T P G P G T P F (In RP 2 the adjugate T is as good as the inverse) A line in L has coordinates L F a b c in F Its coordinates in G are T L F L F P F 0 L F T P G L G L F T Covariant and contravariant tensors � T � for some frame F in RP 2 A point P has coordinates [ P ] F = x y w Let G be the result of transforming F by T 2

(In RP 2 the adjugate T is as good as the inverse) A line in L has coordinates L F a b c in F Its coordinates in G are T L F L F P F 0 L F T P G L G L F T Covariant and contravariant tensors � T � for some frame F in RP 2 A point P has coordinates [ P ] F = x y w Let G be the result of transforming F by T The coordinates [ P ] G of P in G are T ∗ [ P ] F F [ P ] F = G [ P ] G = F T [ P ] G ⇒ [ P ] G = T ∗ [ P ] F 2

A line in L has coordinates L F a b c in F Its coordinates in G are T L F L F P F 0 L F T P G L G L F T Covariant and contravariant tensors � T � for some frame F in RP 2 A point P has coordinates [ P ] F = x y w Let G be the result of transforming F by T The coordinates [ P ] G of P in G are T ∗ [ P ] F F [ P ] F = G [ P ] G = F T [ P ] G ⇒ [ P ] G = T ∗ [ P ] F (In RP 2 the adjugate T ∗ is as good as the inverse) 2

Its coordinates in G are T L F L F P F 0 L F T P G L G L F T Covariant and contravariant tensors � T � for some frame F in RP 2 A point P has coordinates [ P ] F = x y w Let G be the result of transforming F by T The coordinates [ P ] G of P in G are T ∗ [ P ] F F [ P ] F = G [ P ] G = F T [ P ] G ⇒ [ P ] G = T ∗ [ P ] F (In RP 2 the adjugate T ∗ is as good as the inverse) � � A line in L has coordinates [ L ] F = a b c in F 2

Covariant and contravariant tensors � T � for some frame F in RP 2 A point P has coordinates [ P ] F = x y w Let G be the result of transforming F by T The coordinates [ P ] G of P in G are T ∗ [ P ] F F [ P ] F = G [ P ] G = F T [ P ] G ⇒ [ P ] G = T ∗ [ P ] F (In RP 2 the adjugate T ∗ is as good as the inverse) � � A line in L has coordinates [ L ] F = a b c in F Its coordinates in G are T [ L ] F [ L ] F [ P ] F = 0 = [ L ] F T [ P ] G ⇒ [ L ] G = [ L ] F T 2

What we really have is point-like things and line-like things Line-like things “co”-transform with the coordinate system. Point-like things “contra”-transform with the coordinate system. Point-like things are contravariant tensors Line-like (plane-like) things are covariant tensors Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing 3

Line-like things “co”-transform with the coordinate system. Point-like things “contra”-transform with the coordinate system. Point-like things are contravariant tensors Line-like (plane-like) things are covariant tensors Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing What we really have is point-like things and line-like things 3

Point-like things “contra”-transform with the coordinate system. Point-like things are contravariant tensors Line-like (plane-like) things are covariant tensors Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing What we really have is point-like things and line-like things Line-like things “co”-transform with the coordinate system. 3

Point-like things are contravariant tensors Line-like (plane-like) things are covariant tensors Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing What we really have is point-like things and line-like things Line-like things “co”-transform with the coordinate system. Point-like things “contra”-transform with the coordinate system. 3

Line-like (plane-like) things are covariant tensors Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing What we really have is point-like things and line-like things Line-like things “co”-transform with the coordinate system. Point-like things “contra”-transform with the coordinate system. Point-like things are contravariant tensors 3

Covariant and contravariant tensors Lines as row-vectors and points as column vectors are confusing What we really have is point-like things and line-like things Line-like things “co”-transform with the coordinate system. Point-like things “contra”-transform with the coordinate system. Point-like things are contravariant tensors Line-like (plane-like) things are covariant tensors 3

Coordinates of covariant tensors use subscripts L L 1 L 2 L 3 The contraction between a covariant and a contravariant 1-tensor is the scalar product n P i L i P L i 1 Whenever there is an expression with the same index name appearing as a subscript and a subscript, the summation sign is omitted P i L i P 1 L 1 P 2 L 2 P 3 L 3 L 1 P 1 L 2 P 2 L 3 P 3 L i P i Einstein’s notation � P 3 � P 1 P 2 Coordinates of contravariant tensors use superscripts P = 4

The contraction between a covariant and a contravariant 1-tensor is the scalar product n P i L i P L i 1 Whenever there is an expression with the same index name appearing as a subscript and a subscript, the summation sign is omitted P i L i P 1 L 1 P 2 L 2 P 3 L 3 L 1 P 1 L 2 P 2 L 3 P 3 L i P i Einstein’s notation � P 3 � P 1 P 2 Coordinates of contravariant tensors use superscripts P = � � Coordinates of covariant tensors use subscripts L = L 1 L 2 L 3 4

Whenever there is an expression with the same index name appearing as a subscript and a subscript, the summation sign is omitted P i L i P 1 L 1 P 2 L 2 P 3 L 3 L 1 P 1 L 2 P 2 L 3 P 3 L i P i Einstein’s notation � P 3 � P 1 P 2 Coordinates of contravariant tensors use superscripts P = � � Coordinates of covariant tensors use subscripts L = L 1 L 2 L 3 The contraction between a covariant and a contravariant 1-tensor is the scalar product n � P i L i P · L = i = 1 4

Einstein’s notation � P 3 � P 1 P 2 Coordinates of contravariant tensors use superscripts P = � � Coordinates of covariant tensors use subscripts L = L 1 L 2 L 3 The contraction between a covariant and a contravariant 1-tensor is the scalar product n � P i L i P · L = i = 1 Whenever there is an expression with the same index name appearing as a subscript and a subscript, the summation sign is omitted P i L i = P 1 L 1 + P 2 L 2 + P 3 L 3 = L 1 P 1 + L 2 P 2 + L 3 P 3 = L i P i 4

It has two indices, one covariant and one contravariant. It is a mixed 2-tensor M j M j i P i Q j i L j R i The covariant index transforms with T and the contravariant index transforms with T M j i T i N k k T j Transformations A transformation matrix takes a line and returns a line, or takes a point and returns a point 5

It is a mixed 2-tensor M j M j i P i Q j i L j R i The covariant index transforms with T and the contravariant index transforms with T M j i T i N k k T j Transformations A transformation matrix takes a line and returns a line, or takes a point and returns a point It has two indices, one covariant and one contravariant. 5

The covariant index transforms with T and the contravariant index transforms with T M j i T i N k k T j Transformations A transformation matrix takes a line and returns a line, or takes a point and returns a point It has two indices, one covariant and one contravariant. It is a mixed 2-tensor M j i P i = Q j M j i L j = R i 5

Transformations A transformation matrix takes a line and returns a line, or takes a point and returns a point It has two indices, one covariant and one contravariant. It is a mixed 2-tensor M j i P i = Q j M j i L j = R i The covariant index transforms with T and the contravariant index transforms with T ∗ k = M j i T i N ℓ k ( T ∗ ) ℓ j 5

A conic is a purely covariant 2-tensor Q ij P j P i 0 That’s why conics are weird! Both covariant indices transform with T k T j Q ij T i U k Conics Note that in vanilla linear algebra, we only have mixed 2-tensors! 6

That’s why conics are weird! Both covariant indices transform with T k T j Q ij T i U k Conics Note that in vanilla linear algebra, we only have mixed 2-tensors! A conic is a purely covariant 2-tensor Q ij P j P i = 0 6

Conics Note that in vanilla linear algebra, we only have mixed 2-tensors! A conic is a purely covariant 2-tensor Q ij P j P i = 0 That’s why conics are weird! Both covariant indices transform with T k T j U k ℓ = Q ij T i ℓ 6

L connects the tangency points R S of the two tangents to Q through P If P belongs to the conic, L it is the tangent to Q at P Q ij P i Its coordinates are simply L j Proof Q ij R i R j Q ij S i S j 0 and 0 Q ij P i R j Q ij R j P i 0 0 Q ij P i S j Q ij S j P i 0 0 The polar line The polar line L to of a quadric with regard to a point P 7

If P belongs to the conic, L it is the tangent to Q at P Q ij P i Its coordinates are simply L j Proof Q ij R i R j Q ij S i S j 0 and 0 Q ij P i R j Q ij R j P i 0 0 Q ij P i S j Q ij S j P i 0 0 The polar line The polar line L to of a quadric with regard to a point P L connects the tangency points R , S of the two tangents to Q through P 7