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ECM2410: Graphics and Animation
Transformations
Part I
Transformations 1/15
Transformations
Coordinate systems
The reason
Vectors and matrices
The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems
The reason
Vectors and matrices
The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices
The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices
The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates
A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D
The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks
An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Transformations
Coordinate systems The reason
Vectors and matrices The maths
Homogeneous coordinates A trick
Java2D The API
Clocks An example
ReferencesHearn & Baker, Computer Graphics, Chapter 5.Foley et al, Computer Graphics, Chapter 5.
Introduction Transformations 2/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Coordinate Systems
User space
Device space
User space (world coordinates)Space in which parts of scene are modelledMoveable: location set with affineTransform class.
Device space (screen coordinates)Coordinates in which picture is renderedSize set with java.awt.setSize()
Introduction Transformations 3/15
Building Complex Objects
Device Coordinates
Construct complex objects from simpler ones, each havingits own local coordinate system.
Transform objects into place.
Introduction Transformations 4/15
Building Complex Objects
Device Coordinates
Construct complex objects from simpler ones, each havingits own local coordinate system.
Transform objects into place.
Introduction Transformations 4/15
Building Complex Objects
Device Coordinates
Construct complex objects from simpler ones, each havingits own local coordinate system.
Transform objects into place.
Introduction Transformations 4/15
Building Complex Objects
Device Coordinates
Construct complex objects from simpler ones, each havingits own local coordinate system.
Transform objects into place.
Introduction Transformations 4/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
a
a =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
a
a =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
aa =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
aa =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
aa =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
aa =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Coordinates and Points
Cartesian Coordinates
(3,2)
3
2
(0,0)
aa =
(32
)
The Point class represents a 2-dimensional point or vector:
a = new Point2D.Double(3.0, 2.0);double ax = a.getX(); // 3.0double ay = a.getY(); // 2.0
Introduction Transformations 5/15
Matrices
Find[12
]
[1 −3−5 4
]
Matrices Transformations 6/15
Matrices
Find[12
]
[1 −3−5 4
]
Matrices Transformations 6/15
Matrices
Find[12
]
[1 −3−5 4
]
Matrices Transformations 6/15
Matrices
Find[12
] [1 −3−5 4
]
Matrices Transformations 6/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]
=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]
=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]
=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Matrix Transformations
Matrices: operators for transforming vectors
[1 −3−5 4
] [12
]=
[−53
]
Ax = x′
x local coordinatesx′ screen coordinates
Matrices Transformations 7/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Dilbert
public class DilbertTransextends JComponent {
dilbert D = new Dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
D.setColour(Color.black);D.paint(g2); // Screen coordinates
AffineTransform A = new AffineTransform();// Manipulate transformationg2.setTransform(A); // Apply transformD.setColour(Color.magenta);D.paint(g2);
}
Matrices Transformations 8/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]
Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]
Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]
Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]
Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotation Matrices
x′ = Rx[x ′
y ′
]=
[cos θ − sin θsin θ cos θ
] [xy
]
=
[x cos θ − y sin θx sin θ + y cos θ
]Rotation by θ
Rotation matrices leave the length of the vector unchanged.
|x′|2 = (x cos θ − y sin θ)2 + (x sin θ + y cos θ)2
= x2 + y2
= |x|2
Matrices Transformations 9/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Rotations
Dilbert D =new Dilbert("./dilbert.txt");
. . .
AffineTransformation A =new AffineTransformation();
A.rotation(-Math.PI/12);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
Angles in radians:π radians = 180◦
Java2D rotationsincrease clockwise
Matrices Transformations 10/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling
A.scale(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint();
[x ′
y ′
]=
[λxµy
]=
[λ 00 µy
] [xy
]
Diagonal matrices scale thevectorEqual entries on diagonal giveisotropic scalingUnequal entries on diagonal giveanisotropic scalingDilbert was scaled by:A.scale(2.0, 0.5);
Matrices Transformations 11/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Scaling to fit window
public class DilbertAutoscale extends JComponent {private Dilbert D = new dilbert("./dilbert.txt");. . .
public void paint(Graphics g) {Graphics2D g2 = (Graphics2D)g;
GeneralPath path = D.getpath(); // Path describing dilbertRectangle2D bbox = path.getBounds2D(); // Bounding box
AffineTransform A = new AffineTransform();A.scale(0.9*getWidth() / bbox.getMaxX(),
0.9*getHeight() / bbox.getMaxY());g2.transform(A);
D.setColour(Color.blue);D.paint(g2);
}
Exercise:Modify DilbertAutoScale to preserve Dilbert’s aspect ratio.
Matrices Transformations 12/15
Axis Aligned Bounding Box
Smallest axis-parallel rectangle containing the object
Matrices Transformations 13/15
Axis Aligned Bounding Box
Smallest axis-parallel rectangle containing the object
Matrices Transformations 13/15
Axis Aligned Bounding Box
Smallest axis-parallel rectangle containing the object
Matrices Transformations 13/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Shear
A.shear(λ, µ);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + λy
y
]=
[1 λ0 1
] [xy
][
x ′
y ′
]=
[x
µx + y
]=
[1 0µ 1
] [xy
]
One coordinate is leftunchanged.
Dilbert was sheared with[1 10 1
]A.shear(1, 0);
Matrices Transformations 14/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
Translation
A.translate(150.0, 50.0);g2.transform(A);D.setColour(Color.magenta);D.paint(g2);
[x ′
y ′
]=
[x + txy + ty
]
=
[xy
] [txty
]x′ = x + t
Matrices Transformations 15/15
