Orientation of the coordinate system
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- Melvyn Parsons
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1 Orientation of the oordinate system Right-handed oordinate system: -axis by a positive, around the -axis. The -axis is mapped to the i.e., antilokwise, rotation of The -axis is mapped to the -axis by a positive rotation of around the -axis. The -axis is mapped to the -axis by a positive rotation of around the -axis. y x z Computer Graphis: Basi priniples of 3D graphis p.1/61
2 Orientation of the oordinate system Using the thumb of the right hand for the -axis, the forefinger for the -axis and the middle finger for the -axis, one obtains the orret orientation of the oordinate system. A rotation by a positive angle around an oriented axis in the three-dimensional spae refers to an antilokwise rotation for a viewer to whom the axis points. Right-hand rule: When the thumb of the right hand points in the same diretion as the rotation axis and the other fingers form a fist, then the bent fingers indiate the diretion of positive rotation. Computer Graphis: Basi priniples of 3D graphis p.2/61
3 Homogeneous oordinates where represents the point or, more. is represented by where generally, by Computer Graphis: Basi priniples of 3D graphis p.3/61
4 Geometri transformations : Translation by the vetor Translation matrix: Computer Graphis: Basi priniples of 3D graphis p.4/61
5 Geometri transformations : Saling by the fators Saling matrix: Computer Graphis: Basi priniples of 3D graphis p.5/61
6 Geometri transformations -axis by the angle : Computer Graphis: Basi priniples of 3D graphis p.6/61 Rotation around the Rotation matrix:
7 Geometri transformations -axis by the angle : Rotation around the Rotation matrix: Computer Graphis: Basi priniples of 3D graphis p.7/61
8 Geometri transformations -axis by the angle : Computer Graphis: Basi priniples of 3D graphis p.8/61 Rotation around the Rotation matrix:
9 Geometri transformations Rotation around an arbitrary axis by the angle : Shift the rotation by a translation suh that it passed through the origin. -axis, suh that the rotation -plane. Rotation around the axis is mapped to the -axis, suh that the rotation -axis. Rotation around the axis is mapped to the -axis. Rotation by the angle around the Reverse the three first transformations. Computer Graphis: Basi priniples of 3D graphis p.9/61
10 Geometri transformations As already in the ase of 2D graphis, the omposition of transformations an be implemented by matrix multipliation. The last line for all above mentioned matries is. Matrix multipliation preserves this property. In the two-dimensional ase there is exatly one transformation matrix that maps three nonollinear points to three other nonollinear points. In the three-dimensional ase there exists exatly one transformation matrix that maps four nonoplanar points to four other nonoplanar points. Computer Graphis: Basi priniples of 3D graphis p.10/61
11 Geometri transformations Given four nonoplanar points and the target points, the transformation matrix is obtained by solving the system of linear equations (in homogeneous oordinates) where Computer Graphis: Basi priniples of 3D graphis p.11/61
12 Geometri transformations In this sense, transformations an be interpreted as hanging from one oordinate system to another. Computer Graphis: Basi priniples of 3D graphis p.12/61
13 Transformations in Java 3D The lass Transform3D stores 3D transformations as matries (in homogeneous oordinates), analogous to the 2D ase. Transform3D tf = new Transform3D(); generates the identity transformation orresponding to the unit matrix. tf.rotx(theta); defines tf as a rotation by the angle theta around the -axis. roty and rotz analogously for the - and the -axis. tf.set(new AxisAngle4d(x,y,z,theta)); defines a rotation by the angle theta around the. axis in the diretion of the float-vetor Computer Graphis: Basi priniples of 3D graphis p.13/61
14 Transformations in Java 3D is A translation by the float-vetor speified by tf.settranslation(new Vetor3f(x,y,z)); The method tf.setsale(new Vetor3f(x,y,z));. leads to the saling tf.setsale(fator) defines a saling with the same saling fator fator for the -, - and -diretion. Computer Graphis: Basi priniples of 3D graphis p.14/61
15 Transformations in Java 3D Arbitrary transformations an be defined by tf.set(matrix); where matrix is a one-dimensional double-array with 16 values speifying the entries in the matrix. tf.get(matrix); stores the matrix assoiated with the transformation tf in the (one-dimensional) double-array matrix. Composition of transformations (matrix multipliation): tf.mul(tf1,tf2); tf1.mul(tf2); Computer Graphis: Basi priniples of 3D graphis p.15/61
16 Java 3D oordinate system As long as no additional transformations are applied or the viewer s point remains unhanged, the the the -axis in the window points to the right, -axis upward and -axis to the front. Computer Graphis: Basi priniples of 3D graphis p.16/61
17 Elementary geometri objets For 3D objets it is not suffiient to outline their geometry only. A olour or texture must be assigned to the surfae. In addition, refetion properties like shininess must be assigned to the surfae. In Java 3D the lass Appearane is responsible for suh properties of surfaes. The lass Appearane will be disussed in detail in onnetion with illumination. Computer Graphis: Basi priniples of 3D graphis p.17/61
18 Elementary geometri objets For the first objets a very simple Appearane is defined by Appearane myapp = new Appearane(); settomydefaultappearane(myapp, new Color3f(r,g,b)); where the three float-value r,g,b define the olour. The settomydefaultappearane is not a standard method in Java 3D. It was defined for the purposes here. Computer Graphis: Basi priniples of 3D graphis p.18/61
19 Elementary geometri objets Unless mentioned otherwise, real-valued parameters are always given float-values. Box: A box is generated by Box xyzbox = new Box(x,y,z,myApp); The box has the size and is entred in the origin of the oordinate system. Sphere: Sphere rsphere = new Sphere(r,myApp); defines a sphere with radius r and midpoint in the origin of the oordinate system. Computer Graphis: Basi priniples of 3D graphis p.19/61
20 Elementary geometri objets Cylinder: A ylinder with radius r and height h whose entre point is in the origin of the oordinate system is generated by Cylinder rhcyl = new Cylinder(r,h,myApp); Cone: A one with radius r and height h is onstruted by Cone rhcone = new Cone(r,h,myApp); The one is positioned in the same way as the ylinder entred around the -axis with its tip h/2 units above the -plane. Computer Graphis: Basi priniples of 3D graphis p.20/61
21 The senegraph in Java 3D Computer Graphis: Basi priniples of 3D graphis p.21/61
22 The senegraph in Java 3D The geometri objets in the sene (inluding movements and animations) are assigned to the Content Branh. The View Platform (View Branh) defines viewing parameters like the position of the viewer, the diretion of his view or the type of projetion. Lights is responsible for illumination of the sene. The SimpleUniverse provides a simple default View Platform automatially without the need of speifying all the parameters. Computer Graphis: Basi priniples of 3D graphis p.22/61
23 Struture of a Java 3D program import... publi lass MyJava3DClass extends JFrame { publi Canvas3D mycanvas3d; publi MyJava3DClass() {... } publi stati void main(string[] args) { MyJava3DClass myjava3d = new MyJava3DClass(); } } Computer Graphis: Basi priniples of 3D graphis p.23/61
24 publi MyJava3DClass() this.setdefaultcloseoperation(jframe.exit_on_close); mycanvas3d = new Canvas3D( SimpleUniverse.getPreferredConfiguration()); SimpleUniverse simpuniv = new SimpleUniverse(myCanvas3D); simpuniv.getviewingplatform( ).setnominalviewingtransform(); reatesenegraph(simpuniv); addlight(simpuniv); settitle("uebershrift"); setsize(700,700); getcontentpane().add("center", mycanvas3d); setvisible(true); Computer Graphis: Basi priniples of 3D graphis p.24/61
25 Struture of a Java 3D program The method reatesenegraph must be implemented individually for eah program or sene. All geometri objets, information about their surfaes and dynami hanges an be inorporated in this method, representing the ontent branh. The method addlight defines the illumination of the sene and it an also be implemented differently for eah virtual sene. Computer Graphis: Basi priniples of 3D graphis p.25/61
26 Stati ontent branh The ontent branh is strutured in the form of a tree. Its root is usually generated by BranhGroup thesene = new BranhGroup(); The method thesene.addchild( ) an be used to assign nodes either simple objets or more omplex objets in the form of transformation groups to the root. Computer Graphis: Basi priniples of 3D graphis p.26/61
27 StatiSeneExample.java Computer Graphis: Basi priniples of 3D graphis p.27/61
28 The senegraph Computer Graphis: Basi priniples of 3D graphis p.28/61
29 The senegraph Leaf nodes are either elementary geometri objets (abin, tail, rotor, platform, trunk, leaves) or transformations (all nodes beginning with tf) used to position the objets orretly. All inner nodes are transformation groups (identified by the letters tg). Computer Graphis: Basi priniples of 3D graphis p.29/61
30 The senegraph Transformation groups are used to assoiate transformations to elementary geometri objets in order to transform them or to ombine transformation groups to more omplex objets and to onsider and position the ombined objet as a single objet. In this way, the heliopter an be generated in the origin of the oordinate system, it an then be plaed onto the platform and afterwards the platform an be positioned anywhere in the sene together with the heliopter. Computer Graphis: Basi priniples of 3D graphis p.30/61
31 Navigation Adding the ode lines OrbitBehavior ob = new OrbitBehavior(myCanvas3D); ob.setshedulingbounds( new BoundingSphere( new Point3d(0.0,0.0,0.0), Double.MAX_VALUE)); simpuniv.getviewingplatform( ).setviewplatformbehavior(ob); enables navigation through the sene using the mouse. Computer Graphis: Basi priniples of 3D graphis p.31/61
32 Animation (Movements) Instead of a stati transformation to position an objet, a transformation group an also ontain transformations for movements (and also other animations like olour hanges). Interpolators are needed to model the ontinuous transition of an objet or a omplete transformation group from an initial position or state to a final position or state. PositionInterpolator: Linear interpolation between an initial and a final point. (Translation as animation.) Computer Graphis: Basi priniples of 3D graphis p.32/61
33 Interpolators RotationInterpolator: Interpolation of rotating movement. SaleInterpolator: Continuous saling. ColorInterpolator: Continuous hange of a olour. PathInterpolator: Interpolation along various transformations. For example, the PositionPathInterpolator enables a movement along a polyline. Computer Graphis: Basi priniples of 3D graphis p.33/61
34 Alpha values When should the interpolation start? Should the interpolation only go from state zero to state one or should it also be reversed? How long should the transition from state zero to state one take? Should the transition between the two states zero and one be arried out with onstant speed or should it aelerate slowly in the beginning at state zero until a maximum speed is reahed and then slow down again to have a smooth stop in state one? Should the interpolation be arried out just one or should it be repeated? Computer Graphis: Basi priniples of 3D graphis p.34/61
35 Alpha values Alpha a = new Alpha(l,id,tt,pdd,iad, iard,aa1d,dad,dard,aa0d); Computer Graphis: Basi priniples of 3D graphis p.35/61
36 Alpha values l speifies the attribute loopcount: No. of repetitions (for repition without end). id defines the attribute mode: id=alpha.increasing ENABLE: Interpolation from 0 to 1 only. id=alpha.decreasing ENABLE: Interpolation from 1 to 0 only. id=alpha.increasing ENABLE+ Alpha.DECREASING ENABLE: Alternating between interpolation from 0 to 1 and bak to 0. Computer Graphis: Basi priniples of 3D graphis p.36/61
37 Alpha values All following parameters are of type long and speify a duration in milliseonds. tt defines the attribute triggertime: After how many milliseonds after the start of the program Alpha should deliver the first values. pdd defines the attribute phasedelayduration: Alpha remains in the state 0 for phasedelayduration milliseonds after the triggertime has passed. Computer Graphis: Basi priniples of 3D graphis p.37/61
38 Alpha values iad defines the attribute inreasingalphaduration: Length of the transition time from state 0 to state 1. iard defines the attribute inreasingalpharampduration: Duration of the linear aeleration phase until the onstant maximum speed is reahed. Also used for slowing down the movement, before state 1 is reahed, so that the objet omes to a smooth and not a sudden stop. Computer Graphis: Basi priniples of 3D graphis p.38/61
39 Alpha values aa1d defines the attribute alphaatoneduration: Duration for state 1. dad and dard determine the attributes dereasingalphaduration and dereasingalpharampduration, respetively. Same meaning as inreasingalphaduration and inreasingalpharampduration, but for the transition from 1 to 0. aa0d defines the attribute alphaatzeroduration: Duration for state 0. Computer Graphis: Basi priniples of 3D graphis p.39/61
40 PositionInterpolator Definition of the axis on whih the positions to be interpolated lie. Transform3D axis = new Transform3D(); defines the -axis. If a movement along another axis should take plae, the transformation axis must be defined in suh a way that it maps the -axis onto the desired axis. Definition of the orresponding Alpha alpha. Computer Graphis: Basi priniples of 3D graphis p.40/61
41 PositionInterpolator Definition of the PositionInterpolator. PositionInterpolator pi = new PositionInterpolator( alpha,transformgroup,axis, startingpoint,endpoint); transformgroup is the transformation group to whih pi is assigned. startingpoint and endpoint speify the initial and the endpoint on the axis for interpolation. Computer Graphis: Basi priniples of 3D graphis p.41/61
42 PositionInterpolator The (viewing) region in whih the interpolator should be arried out: BoundingSphere bs = new BoundingSphere( new Point3d(0.0,0.0,0.0), Double.MAX_VALUE); pi.setshedulingbounds(bs); For the transformation group transformgroup, hanges must be enabled. transformgroup.setcapability( TransformGroup.ALLOW_TRANSFORM_WRITE); Computer Graphis: Basi priniples of 3D graphis p.42/61
43 Interpolators Finally, pi must be assigned to the transformation group transformgroup. tg.addchild(pi); A RotationsInterpolator is defined in a similar manner. RotationInterpolator ri = new RotationInterpolator( alpha,transformgroup,axis, startangle,endangle); Computer Graphis: Basi priniples of 3D graphis p.43/61
44 Interpolators SaleInterpolator si = new SaleInterpolator( alpha,transformgroup,axis, minsale,maxsale); defines a SaleInterpolator arrying out a ontinuous saling starting with the fator minsale and ending at the fator maxsale. Computer Graphis: Basi priniples of 3D graphis p.44/61
45 Exerpt from the senegraph Computer Graphis: Basi priniples of 3D graphis p.45/61
46 Projetion The projetion of an objet onto a projetion plane is obtained by onneting the points of the objet with the entre of projetion and omputing the intersetion points of these lines, alled projetors, with the projetion plane. The entre of projetion might be at infinity. perspetive projetion parallel projetion Computer Graphis: Basi priniples of 3D graphis p.46/61
47 Projetion For a parallel projetion the entre of projetion is at infinity and the projetors are parallel. If a onrete entre of projetion is given, the projetion is alled perspetive projetion. The entre of projetion represents the position of the viewer. Java3D uses perspetive projetion by default. Swithing to parallel projetion: simpuniv.getviewer().getview(). setprojetionpoliy( View.PARALLEL_PROJECTION); (see ViewParallelProjetion.java) Computer Graphis: Basi priniples of 3D graphis p.47/61
48 Parallel projetion in hom. oord. -plane Example: Projetion plane parallel to the with in homogeneous oordinates: Computer Graphis: Basi priniples of 3D graphis p.48/61
49 Parallel projetion in hom. oord. Applying a suitable geometri transformation, the virtual world oordinate system an also be hanged suh that the projetion plane is parallel to the -plane. Computer Graphis: Basi priniples of 3D graphis p.49/61
50 Persp. projetion in hom. oord. Example: Perspetive projetion with entre of projetion at the origin of the oordinate system and. -plane at projetion plane parallel to the Applying an interept theorem yields y y z z y z 0 x Computer Graphis: Basi priniples of 3D graphis p.50/61
51 Persp. projetion in hom. oord. and i.e. and where the result is in nonnormalised homogeneous oordinates. Computer Graphis: Basi priniples of 3D graphis p.51/61
52 Persp. projetion in hom. oord. Using the -plane as the projetion plane and therefore shifting the entre of projetion to the position on the -axis, results in and In matrix form: Computer Graphis: Basi priniples of 3D graphis p.52/61
53 Persp. projetion in hom. oord. perspetive projetion as a omposition of a geometri transformation and a parallel projetion: This transformation (the right matrix) does not hange the points on the plane. Computer Graphis: Basi priniples of 3D graphis p.53/61
54 Persp. projetion in hom. oord. Given an arbitrary perspetive projetion, its entre an first be plaed in the origin of the oordinate system. Analogous to parallel projetion, suitable rotations map the projetion plane to a plane parallel to the -plane. A translation by the vetor yields a perspetive projetion with its entre of projetion at, whih orresponds to another transformation plus a final parallel projetion to the -plane. Computer Graphis: Basi priniples of 3D graphis p.54/61
55 Persp. projetion in hom. oord. Therefore, any perspetive projetion an be onsidered as a suitable transformation followed by a parallel projetion to the -plane. It is suffiient to onsider the parallel projetion to the -plane in order to understand any other parallel or perspetive projetion. Other projetions an always be onsidered as a transformation plus a final parallel projetion to the -plane. Computer Graphis: Basi priniples of 3D graphis p.55/61
56 Persp. projetion in hom. oord. Changing the position or the diretion of view of the viewer orresponds to applying a suitable transformation before the above mentioned transformations and the parallel projetion. Computer Graphis: Basi priniples of 3D graphis p.56/61
57 Persp. projetion in hom. oord. Properties of the transformation. Consider the point. In homogeneous oordinates: Computer Graphis: Basi priniples of 3D graphis p.57/61
58 Persp. projetion in hom. oord. In artesian oordinates:. Computer Graphis: Basi priniples of 3D graphis p.58/61
59 Persp. projetion in hom. oord. Transformation of the lines passing through the point :. The transformed lines meet in the point are : The lines passing through the point transformed into lines parallel to the -axis.. The transformed lines pass through the point This point is alled vanishing point. Computer Graphis: Basi priniples of 3D graphis p.59/61
60 Persp. projetion in hom. oord. Computer Graphis: Basi priniples of 3D graphis p.60/61
61 Persp. projetion in hom. oord. When the projetion plane intersets with two or three axes of the oordinate system, suh perspetive projetions are alled two-point and three-point perspetive projetion, respetively. Computer Graphis: Basi priniples of 3D graphis p.61/61
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