Summer 2019 Diego Nehab IMPA 1 2D Computer Graphics
Geometry and Transformations
Points defjned by pair of coordinates • Signed distances to perpendicular directed lines • Point where lines cross is the origin Basis of analytic geometry • Connection between Euclidean geometry and algebra • Describe shapes with equations • E.g., lines and circles 2 Cartesian coordinate system
Points defjned by pair of coordinates • Signed distances to perpendicular directed lines • Point where lines cross is the origin Basis of analytic geometry • Connection between Euclidean geometry and algebra • Describe shapes with equations • E.g., lines and circles 2 Cartesian coordinate system
Points defjned by pair of coordinates • Signed distances to perpendicular directed lines • Point where lines cross is the origin Basis of analytic geometry • Connection between Euclidean geometry and algebra • Describe shapes with equations • E.g., lines and circles 2 Cartesian coordinate system
Distance between line and point? Find intersection between line and circle? Find intersection between two circles? Prove that the medians of a triangle are concurrent? 3 Problems
Distance between line and point? Find intersection between line and circle? Find intersection between two circles? Prove that the medians of a triangle are concurrent? 3 Problems
Distance between line and point? Find intersection between line and circle? Find intersection between two circles? Prove that the medians of a triangle are concurrent? 3 Problems
Distance between line and point? Find intersection between line and circle? Find intersection between two circles? Prove that the medians of a triangle are concurrent? 3 Problems
• Defjne sum of vectors v 1 v 2 and multiplication by scalars v 1 v 2 • In R 2 , V is 0 , line through origin, or all of R 2 v 1 v 2 V 2 0 • That spans V v 4 1 0 2 v 1 v 1 2 v 2 1 is l.i. 2 v 2 1 v 1 Set of V of vectors closed by linear combinations • Linear independent set of vectors for V Basis o to each point p p Given origin o , associate vector v V 2 v 2 1 v 1 V Vector Spaces
• In R 2 , V is 0 , line through origin, or all of R 2 v 1 v 2 4 1 2 v 2 1 v 1 v 2 1 V v • That spans V 0 2 2 v 2 0 Set of V of vectors closed by linear combinations 1 v 1 is l.i. • Linear independent set of vectors for V Basis o to each point p p Given origin o , associate vector v Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V
v 1 v 2 4 0 2 v 2 1 v 1 v 2 1 V v • That spans V 0 2 1 2 v 2 Set of V of vectors closed by linear combinations 1 v 1 is l.i. • Linear independent set of vectors for V Basis o to each point p p Given origin o , associate vector v Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V • In R 2 , V is { 0 } , line through origin, or all of R 2
v 1 v 2 4 1 2 v 2 1 v 1 v 2 1 V v • That spans V 0 2 0 Set of V of vectors closed by linear combinations 2 v 2 1 v 1 is l.i. • Linear independent set of vectors for V Basis Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V • In R 2 , V is { 0 } , line through origin, or all of R 2 Given origin o , associate vector v = p − o to each point p
4 2 2 v 2 1 v 1 v 2 1 V v • That spans V 0 1 Set of V of vectors closed by linear combinations 0 2 v 2 1 v 1 is l.i. • Linear independent set of vectors Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V • In R 2 , V is { 0 } , line through origin, or all of R 2 Given origin o , associate vector v = p − o to each point p Basis B = { v 1 , v 2 } for V
4 • That spans V 2 v 2 1 v 1 v 2 1 • Linear independent set of vectors V v Set of V of vectors closed by linear combinations Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V • In R 2 , V is { 0 } , line through origin, or all of R 2 Given origin o , associate vector v = p − o to each point p Basis B = { v 1 , v 2 } for V B is l.i. ⇔ α 1 v 1 + α 2 v 2 = 0 ⇒ α 1 = α 2 = 0
Set of V of vectors closed by linear combinations • Linear independent set of vectors • That spans V 4 Vector Spaces • Defjne sum of vectors v 1 , v 2 and multiplication by scalars α v 1 , v 2 ∈ V ⇒ α 1 v 1 + α 2 v 2 ∈ V • In R 2 , V is { 0 } , line through origin, or all of R 2 Given origin o , associate vector v = p − o to each point p Basis B = { v 1 , v 2 } for V B is l.i. ⇔ α 1 v 1 + α 2 v 2 = 0 ⇒ α 1 = α 2 = 0 v ∈ V ⇔ ∃ α 1 , α 2 | v = α 1 v 1 + α 2 v 2
5 2 v a 11 a 12 a 21 a 22 1 a 11 T v 1 a 21 2 a 21 1 a 22 2 T a 22 Coordinates of a vector in a given basis 1 T v 1 Linear transformations preserve linear combinations T a 21 2 v 2 1 v 1 2 T v 2 Matrix of a linear transformation T a 11 a 12 Linear transformations � � α 1 [ v ] B = ⇔ v = α 1 v 1 + α 2 v 2 α 2
5 v 2 a 22 1 a 21 2 a 21 1 a 11 2 1 a 22 a 21 a 12 a 11 T Coordinates of a vector in a given basis Linear transformations preserve linear combinations a 22 a 21 a 12 a 11 T Matrix of a linear transformation T v Linear transformations � � α 1 [ v ] B = ⇔ v = α 1 v 1 + α 2 v 2 α 2 T ( α 1 v 1 + α 2 v 2 ) = α 1 T ( v 1 ) + α 2 T ( v 2 )
5 a 11 a 11 a 12 a 21 a 22 1 2 1 T a 21 2 a 21 1 a 22 2 v T v Coordinates of a vector in a given basis Matrix of a linear transformation a 22 a 21 a 12 a 11 Linear transformations preserve linear combinations Linear transformations � � α 1 [ v ] B = ⇔ v = α 1 v 1 + α 2 v 2 α 2 T ( α 1 v 1 + α 2 v 2 ) = α 1 T ( v 1 ) + α 2 T ( v 2 ) � � [ T ] B =
5 a 11 a 22 a 21 a 12 a 11 Coordinates of a vector in a given basis a 22 Linear transformations preserve linear combinations a 21 Matrix of a linear transformation a 12 Linear transformations � � α 1 [ v ] B = ⇔ v = α 1 v 1 + α 2 v 2 α 2 T ( α 1 v 1 + α 2 v 2 ) = α 1 T ( v 1 ) + α 2 T ( v 2 ) � � [ T ] B = � � � � � � α 1 a 11 α 1 + a 21 α 2 [ T ( v )] B = [ T ] B [ v ] B = = α 2 a 21 α 1 + a 22 α 2
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
Interesting transformations • Identity, Rotation, Scale, Refmection, Shearing • Scale along arbitrary direction • No translation. Why? [Klein] A Geometry is the set of properties preserved by a group of transformations General linear group • Composition, inverse • Preserves collinearity, parallelism, concurrency, tangency, ratios of distances along lines 6 Linear transformations
v 2 v 2 7 v u u v v u v y u y v u v x u x with the x-axis and Let u and v make angles u u Dot product, scalar product, standard inner product u 2 u v uov Euclydean norm, or vector length u x 4 u 2 y u u Conversely u v 1 Norm and inner product u T v = u · v = � u , v � = u x v x + u y v y
v 2 v 2 u v x u Let u and v make angles and with the x-axis u x 7 u u y u v y v u v u v v 4 Dot product, scalar product, standard inner product 1 u v uov Euclydean norm, or vector length u 2 Conversely u v Norm and inner product u T v = u · v = � u , v � = u x v x + u y v y � � � u � = x + u 2 y = � u , u �
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