2D Computer Graphics Geometry and Transformations Points defjned by - - PowerPoint PPT Presentation

2D Computer Graphics

Diego Nehab Summer 2019

IMPA 1

Geometry and Transformations

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

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

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

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

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1 v2 and multiplication by scalars

v1 v2 V

1v1 2v2

V

• In R2, V is 0 , line through origin, or all of R2

Given origin o, associate vector v p

• to each point p

Basis v1 v2 for V

• Linear independent set of vectors

is l.i.

1v1 2v2 1 2

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is 0 , line through origin, or all of R2

Given origin o, associate vector v p

• to each point p

Basis v1 v2 for V

• Linear independent set of vectors

is l.i.

1v1 2v2 1 2

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is {0}, line through origin, or all of R2

Given origin o, associate vector v p

• to each point p

Basis v1 v2 for V

• Linear independent set of vectors

is l.i.

1v1 2v2 1 2

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is {0}, line through origin, or all of R2

Given origin o, associate vector v = p − o to each point p Basis v1 v2 for V

• Linear independent set of vectors

is l.i.

1v1 2v2 1 2

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is {0}, line through origin, or all of R2

Given origin o, associate vector v = p − o to each point p Basis B = {v1, v2} for V

• Linear independent set of vectors

is l.i.

1v1 2v2 1 2

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is {0}, line through origin, or all of R2

Given origin o, associate vector v = p − o to each point p Basis B = {v1, v2} for V

• Linear independent set of vectors

B is l.i. ⇔ α1v1 + α2v2 = 0 ⇒ α1 = α2 = 0

• That spans V

v V

1 2

v

1v1 2v2 4

Vector Spaces

Set of V of vectors closed by linear combinations

• Defjne sum of vectors v1, v2 and multiplication by scalars α

v1, v2 ∈ V ⇒ α1v1 + α2v2 ∈ V

• In R2, V is {0}, line through origin, or all of R2

Given origin o, associate vector v = p − o to each point p Basis B = {v1, v2} for V

• Linear independent set of vectors

B is l.i. ⇔ α1v1 + α2v2 = 0 ⇒ α1 = α2 = 0

• That spans V

v ∈ V ⇔ ∃ α1, α2 | v = α1v1 + α2v2

4

Linear transformations

Coordinates of a vector in a given basis [v]B =

• α1

α2

• ⇔ v = α1v1 + α2v2

Linear transformations preserve linear combinations T

1v1 2v2 1T v1 2T v2

Matrix of a linear transformation T a11 a12 a21 a22 T v T v a11 a12 a21 a22

1 2

a11

1

a21

2

a21

1

a22

2 5

Linear transformations

Coordinates of a vector in a given basis [v]B =

• α1

α2

• ⇔ v = α1v1 + α2v2

Linear transformations preserve linear combinations T(α1v1 + α2v2) = α1T(v1) + α2T(v2) Matrix of a linear transformation T a11 a12 a21 a22 T v T v a11 a12 a21 a22

1 2

a11

1

a21

2

a21

1

a22

2 5

Linear transformations

Coordinates of a vector in a given basis [v]B =

• α1

α2

• ⇔ v = α1v1 + α2v2

Linear transformations preserve linear combinations T(α1v1 + α2v2) = α1T(v1) + α2T(v2) Matrix of a linear transformation [T]B =

• a11

a12 a21 a22

• T v

T v a11 a12 a21 a22

1 2

a11

1

a21

2

a21

1

a22

2 5

Linear transformations

Coordinates of a vector in a given basis [v]B =

• α1

α2

• ⇔ v = α1v1 + α2v2

Linear transformations preserve linear combinations T(α1v1 + α2v2) = α1T(v1) + α2T(v2) Matrix of a linear transformation [T]B =

• a11

a12 a21 a22

• [T(v)]B = [T]B[v]B =
• a11

a12 a21 a22 α1 α2

• =
• a11α1 + a21α2

a21α1 + a22α2

• 5

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

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

Norm and inner product

Dot product, scalar product, standard inner product uTv = u · v = u, v = uxvx + uyvy u v uov Euclydean norm, or vector length u u2

x

u2

y

u u Conversely u v 1 4 u v 2 u v 2 Let u and v make angles and with the x-axis ux u vx v uy u vy v u v u v

7

Norm and inner product

Dot product, scalar product, standard inner product uTv = u · v = u, v = uxvx + uyvy u v uov Euclydean norm, or vector length u =

• u2

x + u2 y =

• u, u

Conversely u v 1 4 u v 2 u v 2 Let u and v make angles and with the x-axis ux u vx v uy u vy v u v u v

7

Norm and inner product

Dot product, scalar product, standard inner product uTv = u · v = u, v = uxvx + uyvy u v uov Euclydean norm, or vector length u =

• u2

x + u2 y =

• u, u

Conversely u, v = 1 4(u + v2 − u − v2) Let u and v make angles and with the x-axis ux u vx v uy u vy v u v u v

7

Norm and inner product

Dot product, scalar product, standard inner product uTv = u · v = u, v = uxvx + uyvy u v uov Euclydean norm, or vector length u =

• u2

x + u2 y =

• u, u

Conversely u, v = 1 4(u + v2 − u − v2) Let u and v make angles α and β with the x-axis cos(β − α) = cos(α) cos(β) + sin(α) sin(β) = ux/uvx/v + uy/uvy/v = u, v/(uv)

7

Norm and inner product

Dot product, scalar product, standard inner product uTv = u · v = u, v = uxvx + uyvy = uv cos(∠uov) Euclydean norm, or vector length u =

• u2

x + u2 y =

• u, u

Conversely u, v = 1 4(u + v2 − u − v2) Let u and v make angles α and β with the x-axis cos(β − α) = cos(α) cos(β) + sin(α) sin(β) = ux/uvx/v + uy/uvy/v = u, v/(uv)

7

Euclidean Geometry

Orthogonal transformations

• Inverse is transpose, or equivalently
• Preserve inner products

Euclidean group

• Rigid transformations (isometries), or
• Orthogonal transformation and translations
• Preserves collinearity, parallelism, angles, concurrency, tangency,

distance between points How to represent?

8

Euclidean Geometry

Orthogonal transformations

• Inverse is transpose, or equivalently
• Preserve inner products

Euclidean group

• Rigid transformations (isometries), or
• Orthogonal transformation and translations
• Preserves collinearity, parallelism, angles, concurrency, tangency,

distance between points How to represent?

8

Euclidean Geometry

Orthogonal transformations

• Inverse is transpose, or equivalently
• Preserve inner products

Euclidean group

• Rigid transformations (isometries), or
• Orthogonal transformation and translations
• Preserves collinearity, parallelism, angles, concurrency, tangency,

distance between points How to represent?

8

Euclidean Geometry

Orthogonal transformations

• Inverse is transpose, or equivalently
• Preserve inner products

Euclidean group

• Rigid transformations (isometries), or
• Orthogonal transformation and translations
• Preserves collinearity, parallelism, angles, concurrency, tangency,

distance between points How to represent?

8

Euclidean Geometry

Orthogonal transformations

• Inverse is transpose, or equivalently
• Preserve inner products

Euclidean group

• Rigid transformations (isometries), or
• Orthogonal transformation and translations
• Preserves collinearity, parallelism, angles, concurrency, tangency,

distance between points How to represent?

8

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Similarity geometry

Few properties are exclusive to Euclidean geometry Similarity group

• Rigid transformations and uniform scale (dilation)
• Does not preserve distances
• Maps between any two pairs of points
• Preserves collinearity, parallelism, angles, concurrency, tangency

How to represent?

9

Affine spaces

Useful to represent solutions to a linear system Also useful represent translations Let V be a vector space with basis v1 v2 and o a point Affjne space is A

• V

p p

• V
• Affjne frame

v1 v2 o p

• 1v1

2v2

• Affjne coordinates

p

1 2

1

10

Affine spaces

Useful to represent solutions to a linear system Also useful represent translations Let V be a vector space with basis v1 v2 and o a point Affjne space is A

• V

p p

• V
• Affjne frame

v1 v2 o p

• 1v1

2v2

• Affjne coordinates

p

1 2

1

10

Affine spaces

Useful to represent solutions to a linear system Also useful represent translations Let V be a vector space with basis B = {v1, v2} and o a point Affjne space is A = o + V = {p | p − o ∈ V}

• Affjne frame

v1 v2 o p

• 1v1

2v2

• Affjne coordinates

p

1 2

1

10

Affine spaces

Useful to represent solutions to a linear system Also useful represent translations Let V be a vector space with basis B = {v1, v2} and o a point Affjne space is A = o + V = {p | p − o ∈ V}

• Affjne frame C = {v1, v2; o}

p = o + α1v1 + α2v2

• Affjne coordinates

p

1 2

1

10

Affine spaces

Useful to represent solutions to a linear system Also useful represent translations Let V be a vector space with basis B = {v1, v2} and o a point Affjne space is A = o + V = {p | p − o ∈ V}

• Affjne frame C = {v1, v2; o}

p = o + α1v1 + α2v2

• Affjne coordinates

[p]C =    α1 α2 1   

10

Affine spaces

Let V be a vector space with basis B = {v1, v2} and o a point Barycentric frame D = {a0, a1, a2} = {o, o + v1, o + v2} p

• 1v1

2v2

1

1 2 o 1 o

v1

2 o

v2 1

1 2 a0 1a1 2a2 0a0 1a1 2a2

with

1 2

1

• Barycentric coordinates p

1 2

• Displacement vectors v

V are such that

2 i i

• Points p

A are such that

2 i i

1 (affjne combination)

11

Affine spaces

Let V be a vector space with basis B = {v1, v2} and o a point Barycentric frame D = {a0, a1, a2} = {o, o + v1, o + v2} p = o + α1v1 + α2v2 = (1 − α1 − α2)o + α1(o + v1) + α2(o + v2) = (1 − α1 − α2)a0 + α1a1 + α2a2 = α0a0 + α1a1 + α2a2, with α0 + α1 + α2 = 1.

• Barycentric coordinates p

1 2

• Displacement vectors v

V are such that

2 i i

• Points p

A are such that

2 i i

1 (affjne combination)

11

Affine spaces

Let V be a vector space with basis B = {v1, v2} and o a point Barycentric frame D = {a0, a1, a2} = {o, o + v1, o + v2} p = o + α1v1 + α2v2 = (1 − α1 − α2)o + α1(o + v1) + α2(o + v2) = (1 − α1 − α2)a0 + α1a1 + α2a2 = α0a0 + α1a1 + α2a2, with α0 + α1 + α2 = 1.

• Barycentric coordinates [p]D =

   α0 α1 α2   

• Displacement vectors v

V are such that

2 i i

• Points p

A are such that

2 i i

1 (affjne combination)

11

Affine spaces

Let V be a vector space with basis B = {v1, v2} and o a point Barycentric frame D = {a0, a1, a2} = {o, o + v1, o + v2} p = o + α1v1 + α2v2 = (1 − α1 − α2)o + α1(o + v1) + α2(o + v2) = (1 − α1 − α2)a0 + α1a1 + α2a2 = α0a0 + α1a1 + α2a2, with α0 + α1 + α2 = 1.

• Barycentric coordinates [p]D =

   α0 α1 α2   

• Displacement vectors v ∈ V are such that 2

i=0 αi = 0

• Points p

A are such that

2 i i

1 (affjne combination)

11

Affine spaces

Let V be a vector space with basis B = {v1, v2} and o a point Barycentric frame D = {a0, a1, a2} = {o, o + v1, o + v2} p = o + α1v1 + α2v2 = (1 − α1 − α2)o + α1(o + v1) + α2(o + v2) = (1 − α1 − α2)a0 + α1a1 + α2a2 = α0a0 + α1a1 + α2a2, with α0 + α1 + α2 = 1.

• Barycentric coordinates [p]D =

   α0 α1 α2   

• Displacement vectors v ∈ V are such that 2

i=0 αi = 0

• Points p ∈ A are such that 2

i=0 αi = 1 (affjne combination) 11

Affine transformations

Combination of two arbitrary parallel projections Preserve affjne combinations

1 2

1 T

0a0 1a1 2a2 0T a0 1T a1 2T a2

Matrix of an affjne transformation in affjne frame T a11 a12 t1 a21 a22 t2 1 T p T p a11 a12 t1 a21 a22 t2 1

1 2

1 a11

1

a21

2

t1 a21

1

a22

2

t2 1

12

Affine transformations

Combination of two arbitrary parallel projections Preserve affjne combinations α0 + α1 + α2 = 1 ⇒ T(α0a0 + α1a1 + α2a2) = α0T(a0) + α1T(a1) + α2T(a2) Matrix of an affjne transformation in affjne frame T a11 a12 t1 a21 a22 t2 1 T p T p a11 a12 t1 a21 a22 t2 1

1 2

1 a11

1

a21

2

t1 a21

1

a22

2

t2 1

12

Affine transformations

Combination of two arbitrary parallel projections Preserve affjne combinations α0 + α1 + α2 = 1 ⇒ T(α0a0 + α1a1 + α2a2) = α0T(a0) + α1T(a1) + α2T(a2) Matrix of an affjne transformation in affjne frame [T]C =    a11 a12 t1 a21 a22 t2 1    T p T p a11 a12 t1 a21 a22 t2 1

1 2

1 a11

1

a21

2

t1 a21

1

a22

2

t2 1

12

Affine transformations

Combination of two arbitrary parallel projections Preserve affjne combinations α0 + α1 + α2 = 1 ⇒ T(α0a0 + α1a1 + α2a2) = α0T(a0) + α1T(a1) + α2T(a2) Matrix of an affjne transformation in affjne frame [T]C =    a11 a12 t1 a21 a22 t2 1    [T(p)]C = [T]C[p]C =    a11 a12 t1 a21 a22 t2 1       α1 α2 1    =    a11α1 + a21α2 + t1 a21α1 + a22α2 + t2 1   

12

Affine transformations

Interesting transformations

• Translation, rotation, scale
• Centered rotation
• Centered scale
• Scale in arbitrary direction

What about the matrix in barycentric frame a0 a1 a2

13

Affine transformations

Interesting transformations

• Translation, rotation, scale
• Centered rotation
• Centered scale
• Scale in arbitrary direction

What about the matrix in barycentric frame a0 a1 a2

13

Affine transformations

Interesting transformations

• Translation, rotation, scale
• Centered rotation
• Centered scale
• Scale in arbitrary direction

What about the matrix in barycentric frame a0 a1 a2

13

Affine transformations

Interesting transformations

• Translation, rotation, scale
• Centered rotation
• Centered scale
• Scale in arbitrary direction

What about the matrix in barycentric frame a0 a1 a2

13

Affine transformations

Interesting transformations

• Translation, rotation, scale
• Centered rotation
• Centered scale
• Scale in arbitrary direction

What about the matrix in barycentric frame D = {a0, a1, a2}

13

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Affine geometry

What remains of Euclidean geometry when we forget about distance and angle Affjne group

• Non-singular linear transformation and translation
• Maps between any two sets of three non-collinear points
• Preserves collinearity, parallelism, concurrency, tangency, ratios of

distances along parallel lines There is only one triangle, ellipse, parabola, and hyperbola Visualization of the affjne plane

14

Lines and conics

Line ax + by + c = 0 nTp = 0, with nT =

• a

b c

• and

p =    x y 1   

• How does it change with an affjne transformation?

Conic ax2 2bxy cy2 2dx 2ey f pTC p with C CT a b d b c e d e f and p x y 1

• How does it change with an affjne transformation?

15

Lines and conics

Line ax + by + c = 0 nTp = 0, with nT =

• a

b c

• and

p =    x y 1   

• How does it change with an affjne transformation?

Conic ax2 2bxy cy2 2dx 2ey f pTC p with C CT a b d b c e d e f and p x y 1

• How does it change with an affjne transformation?

15

Lines and conics

Line ax + by + c = 0 nTp = 0, with nT =

• a

b c

• and

p =    x y 1   

• How does it change with an affjne transformation?

Conic ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0 pTC p = 0, with C = CT =    a b d b c e d e f    and p =    x y 1   

• How does it change with an affjne transformation?

15

Lines and conics

Line ax + by + c = 0 nTp = 0, with nT =

• a

b c

• and

p =    x y 1   

• How does it change with an affjne transformation?

Conic ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0 pTC p = 0, with C = CT =    a b d b c e d e f    and p =    x y 1   

• How does it change with an affjne transformation?

15

Revisiting 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?

16

RP2

Projective points: lines through origin in 3D

• Ideal points

Projective lines: planes through origin in 3D

• Ideal line or line at infjnity

Projective plane

• Affjne plane augmented with ideal points

Homogeneous coordinates

• Generalization of affjne coordinates

a b c a b c not all zero w x w y w x y 1 w

17

RP2

Projective points: lines through origin in 3D

• Ideal points

Projective lines: planes through origin in 3D

• Ideal line or line at infjnity

Projective plane

• Affjne plane augmented with ideal points

Homogeneous coordinates

• Generalization of affjne coordinates

a b c a b c not all zero w x w y w x y 1 w

17

RP2

Projective points: lines through origin in 3D

• Ideal points

Projective lines: planes through origin in 3D

• Ideal line or line at infjnity

Projective plane

• Affjne plane augmented with ideal points

Homogeneous coordinates

• Generalization of affjne coordinates

a b c a b c not all zero w x w y w x y 1 w

17

RP2

Projective points: lines through origin in 3D

• Ideal points

Projective lines: planes through origin in 3D

• Ideal line or line at infjnity

Projective plane

• Affjne plane augmented with ideal points

Homogeneous coordinates

• Generalization of affjne coordinates

   a b c    , a, b, c not all zero    w x w y w    ≡    x y 1    , w = 0

17

Projective transformations

Combination of three arbitrary perspective transformations Matrix of a projective transformation T a11 a12 a13 a21 a22 a23 a31 a32 a33 a11 a12 a13 a21 a22 a23 a31 a32 a33

1 2 3

a11

1

a21

2

a13

3

a21

1

a22

2

a23

3

a31

1

a32

2

a33

3

Must be invertible

18

Projective transformations

Combination of three arbitrary perspective transformations Matrix of a projective transformation [T] =    a11 a12 a13 a21 a22 a23 a31 a32 a33       a11 a12 a13 a21 a22 a23 a31 a32 a33       α1 α2 α3    =    a11α1 + a21α2 + a13α3 a21α1 + a22α2 + a23α3 a31α1 + a32α2 + a33α3    Must be invertible

18

Projective transformations

Combination of three arbitrary perspective transformations Matrix of a projective transformation [T] =    a11 a12 a13 a21 a22 a23 a31 a32 a33       a11 a12 a13 a21 a22 a23 a31 a32 a33       α1 α2 α3    =    a11α1 + a21α2 + a13α3 a21α1 + a22α2 + a23α3 a31α1 + a32α2 + a33α3    Must be invertible

18

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

Projective geometry

Projective linear group

• Non-singular linear transformations in R3
• Preserves collinearity, tangency, cross-ratios
• Maps between any two sets of 4 points non-collinear 3 by 3

All lines meet, even parallel lines All quadrilaterals are the same All conics are the same

19

References

• D. A. Brannan, M. F. Esplen, and J. J. Gray. Geometry. Cambridge

University Press, 2011.

20