Chapter5 Transformations

Computer GraphicsChapter 5Geometric TransformationsAndreas Savva

2D TranslationRepositioning an object along a straight line path from one co-ordinate location to another(x,y) (x,y) To translate a 2D position, we add translation distances tx and ty to the original coordinates (x,y) to obtain the new coordinate position (x,y)x= x + tx , y= y + tyMatrix form

2D TranslationMoving a polygon from position (a) to position (b) with the translation vector (-5, 10), i.e.(a)(b)

Translating a Polygonclass Point2D { public: GLfloat x, y;};

void translatePoly(Point2D P[], GLint n, GLfloat tx, GLfloat ty){GLint i;for (i=0; i

2D RotationRepositioning an object along a circular path in the xy-planeThe original coordinates are:

2D RotationSubstitutingMatrix form

2D Rotation about a Pivot positionRotating about pivot position (xr, yr)

2D ScalingAltering the size of an object. Sx and Sy are the scaling factors. If Sx = Sy then uniform scaling.Matrix form

2D Scaling relative to Fixed pointScaling relative to fixed point (xf, yf)ORwhere the additive terms xf(1-Sx) and yf(1-Sy) areconstants for all points in the object.

Matrix RepresentationUse 33 matrices to combine transformations

Translation

Rotation

Scaling

Inverse TransformationsTranslation

Rotation

Scaling

ExampleConsider the line with endpoints (10, 10) and (30, 25). Translate it by tx = -20, ty = -10 and then rotate it by = 90.Right-to-left

Solution

Solution (continue)Point (10, 10)Point (30, 25)

ResultStep-by-stepT(-20, -10)R(90)

ExercisesConsider the following object:

Apply a rotation by 145 then scale it by Sx=2 and Sy=1.5 and then translate it by tx=20 and ty=-30.Scale it by Sx= and Sy=2 and then rotate it by 30.Apply a rotation by 90 and then another rotation by 45.Apply a rotation by 135.

ExercisesComposite 2D TransformationsTranslation: Show that:

Rotation: Show that:

Scaling: Show that:

General 2D Pivot-Point RotationOriginal positionand Pivot PointTranslate Object so thatPivot Point is at originRotation about originTranslate object so that Pivot Pointis return to position (xr , yr)

General Pivot-point RotationUsing Matrices

ExercisesConsider the following object:

Apply a rotation by 60 on the Pivot Point (-10, 10) and display it.Apply a rotation by 30 on the Pivot Point (45, 10) and display it.Apply a rotation by 270 on the Pivot Point (10, 0) and then translate it by tx = -20 and ty = 5. Display the final result.

General 2D Fixed-Point ScalingOriginal positionand Fixed PointTranslate Object so thatFixed Point is at originScale Object withrespect to originTranslate Object so that Fixed Pointis return to position (xf , yf)

General 2D Fixed-Point ScalingUsing Matrices

ExercisesConsider the following object:

Scale it by sx = 2 and sy = relative to the fixed point (140, 125) and display it.Apply a rotation by 90 on the Pivot Point (50, 60) and then scale it by sx = sy = 2 relative to the Fixed Point (0, 200). Display the result.Scale it sx = sy = relative to the Fixed Point (50, 60) and then rotate it by 180 on the Pivot Point (50, 60). Display the final result.

Order of Transformations

ReflectionAbout the x axisAbout the y axis

ReflectionRelative to the coordinate originWith respect to the line y = x

2D Shearx-direction shearMatrix form

2D Shearx-direction relative to other reference lineMatrix formy

2D Sheary-direction shearMatrix form

2D Sheary-direction relative to other reference lineMatrix formshy = , xref = -1

Transformations between 2D Coordinate SystemsTo translate object descriptions from xy coordinates to xy coordinates, we set up a transformation that superimposes the xy axes onto the xy axes. This is done in two steps:Translate so that the origin (x0, y0) of the xy system is moved to the origin (0, 0) of the xy system.Rotate the x axis onto the x axis.

Transformations between 2DCoordinate Systemsi.e.1)

2)

Concatenating:

ExampleFind the xy-coordinates of the xy points (10, 20) and (35, 20), as shown in the figure below:

ExerciseFind the xy-coordinates of the rectangle shown in the figure below:

3D TranslationRepositioning an object along a straight line path from one co-ordinate location to another(x,y,z) (x,y,z) To translate a 3D position, we add translation distances tx ty and tz to the original coordinates (x,y,z) to obtain the new coordinate position (x,y)x= x + tx , y= y + ty , z= z + tz

Matrix form (4 4)

3D Rotationz-axisThe 2D z-axis rotation equations are extended to 3D.Matrix form

3D Rotationx-axisMatrix form

3D Rotationy-axisMatrix form

3D ScalingMatrix form

Other 3D TransformationsReflection z-axisShears z-axis