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3D Graphics : Computer Graphics Fundamentals
The document covers 3D graphics and transformations, including methods like translation, rotation, scaling, reflection, and shearing using matrix representations. It explains the display of 3D objects on a 2D plane through techniques such as parallel and perspective projection, depth cueing, and surface rendering. Additionally, it discusses object representation, including boundary representations, polygon meshes, and spline representations, concluding with the specification of splines and Bezier curves for modeling complex shapes.
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Overview of 3D graphics concept.
Covers geometric transformations including translation, rotation, scaling, reflection, and shear with matrix representations.
Discusses methods for generating and displaying 3D objects in 2D space with camera coordinate setups.
Explains parallel and perspective projection techniques for visual representation of 3D objects.
Methods for depth cueing, surface rendering, and the visual identification of surfaces in 3D graphics.
Explores exploded views, cutaway views, and stereoscopic presentations for enhanced visualization.
Describes various representations of 3D objects including polygons, splines, and procedural methods.
Details on polygon representations, including geometric and attribute data tables for 3D modeling.
Describes equations for plane surfaces using coordinates and the position of points relative to surfaces.
Discusses polygon meshes and types of quadric surfaces like spheres and torus in 3D modeling.
Introduces superquadrics, which allow for flexible shape adjustments in 3D graphics.
Explains spline curves, control points, and conditions for parametric continuity within curves.
Details on Bezier curves, their properties, and applications in surface designs using control points.
Describes B-spline curves and surfaces allowing local control for shaping and modifying features.
Discusses octree structures for representing solid objects, including their use in medical applications.
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3D Graphics : Computer Graphics Fundamentals
- 1.
- 2. 3D Transformations Geometrictransformations and object modeling in 3D are extended from 2D methods by including considerations for the z coordinate. Translation A point is translated from position P=(x,y, z) to position P’=(x’,y’, z’) with matrix operation 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 x y z x t x y t y z t z x y z x x t y y t z z t , ,x y z , ,x y z x y z
- 3. Rotation Positive rotationangles produce anticlockwise rotations about a coordinate axis
- 4.
- 6. Scaling The matrixexpression for the scaling transformation of a position P = (x, y, z) relative to coordinate origin can be written as 11000 000 000 000 1 ' ' ' z y x s s s z y x z y x x y z x y z z ' ' ' .zz . . s syy sxx y x
- 7. The matrixrepresentation for an arbitrary fixed-point (xf, yf, zf) can be expressed as: 1000 )1(00 )1(00 )1(00 ),,(),,(),,( fzz fyy fxx fffzyxfff zss yss xss zyxTsssSzyxT
- 8. Reflections The matrixexpression for the reflection transformation of a position P = (x, y, z) relative to xy plane can be written as: similarly, as reflections relative to yz plane and xz plane, respectively.
- 9. Shear The matrixexpression for the shearing transformation of a position P = (x, y, z) Transformation in z axis 1000 0100 010 001 y x z sh sh SH
- 10. Transformation in yaxis Transformation in x axis 1000 010 0010 001 z x y sh sh SH 1000 010 001 0001 z y x sh sh SH
- 11. 3D Display Methods 3D graphics deals with generating and displaying three dimensional objects in a two-dimensional space(eg: display screen). In addition to color and brightness, a 3-D pixels adds a depth property that indicates where the point lies on the imaginary z-axis. To generate realistic picture we have to first setup a coordinate reference for camera. This co-ordinate reference defines the position and orientation for the plane of the camera.
- 12. This planeused to display a view of the object Object description has to transfer to the camera reference co- ordinates and projected onto the selected display plane.
- 13. Parallel Projection Project pointson the object surface along parallel lines onto the display plane. Parallel lines are still parallel after projection. Used in engineering and architectural drawings. Views maintain relative proportions of the object. Top View Side View Front View
- 14. Perspective Projection • Projectpoints to the display plane along converging paths. • This is the way that our eyes and a camera lens form images and so the displays are more realistic. • Parallel lines appear to converge to a distant point in the background. • Distant objects appear smaller than objects closer to the viewing position.
- 15. Depth Cueing To easilyidentify the front and back of display objects. Depth information can be included using various methods. A simple method to vary the intensity of objects according to their distance from the viewing position. Eg: lines closest to the viewing position are displayed with the higher intensities and lines farther away are displayed with lower intensities.
- 16. Application :modelingthe effect of the atmosphere on the pixel intensity of objects. More distant objects appear dimmer to us than nearer objects due to light scattering by dust particles, smoke etc.
- 17. Visible line andsurface identification • Highlight the visible lines or display them in different color • Display nonvisible lines as dashed lines • Remove the nonvisible lines
- 18. Surface rendering • Setthe surface intensity of objects according to Lighting conditions in the scene Assigned surface characteristics Lighting specifications include the intensity and positions of light sources and the general background illumination required for a scene. Surface properties include degree of transparency and how rough or smooth of the surfaces
- 19. Exploded and CutawayViews To maintain a hierarchical structures to include internal details. These views show the internal structure and relationships of the object parts
- 20. Stereoscopic Views Todisplay objects using stereoscopic views Stereoscopic devices present 2 views of scene One for left eye. Other for right eye. These two views displayed on alternate refresh cycle of a raster monitor Then viewed through glasses that alternately darken first one lens then the other in synchronized with the monitor refresh cycle.
- 21. 3D Object Representation Graphics scenes contain many different kinds of objects and material surfaces Trees, flowers, clouds, rocks, water, bricks, wood paneling, rubber, paper, steel, glass, plastic and cloth Polygon and Quadric surfaces: For simple Euclidean objects eg: polyhedron and ellipsoid Spline surfaces and construction: For curved surfaces eg: aircraft wings , gears Procedural methods – Fractals: For natural objects eg: cloud, grass Octree Encoding: For internal features of objects eg:CT image
- 22. Representation schemes categoriesinto 2 Boundary representation(B –reps) A set of surfaces that separate the object interior from the environment Eg) Polygon facets, spline patches Space-partitioning representation Used to describe interior properties. Partitioning the spatial region into a set of small, non overlapping, contiguous solids (usually cubes) Eg) octree representation
- 23. Polygon Surfaces Mostcommonly used boundary representation. Polygon table Specify a polygon surfaces using vertex coordinates and attribute parameter. Polygon data table organized into 2 group. 1. Geometric data table: vertex coordinate and parameter to identify the spatial orientation. 3 lists Vertex table –coordinate values of each vertex. Edge table - pointer back to vertex table to identify the vertices for polygon edge. Polygon table- pointer back to edge table to identify the edges for each polygon
- 25. 2. Attribute table:Degree of transparency and surface reflectivity etc. Some consistency checks of the geometric data table: Every vertex is listed as an endpoint for at least 2 edges. Every edge is part of at least one polygon. Every polygon is closed. Each polygon has at least one shared edge.
- 26. Plane Equation Theequation for a plane surface expressed at the form Ax+By+Cz+D=0 We can obtain the values of A,B,C,D by solving 3 plane equations using the coordinate values for 3 noncollinear points in the plane(x1,y1, z1), (x2,y2, z2), (x3,y3, z3). 3,2,1,1z)/()/()/( k kDCyDBxDA kk
- 28. If we substituteany arbitrary point (x,y, z) into this equation, then, Ax + By + Cz + D ≠0 implies that the point (x,y,z) not on a surface Ax + By + Cz + D < 0 implies that the point (x,y,z) is inside the surface. Ax + By + Cz + D >0 implies that the point (x,y,z) is outside the surface.
- 29. Polygon Meshes Objectsurfaces are tiled to specify the surface facets with mesh function. Triangle strip - Produce n-2 connected triangles ,for n vertices Quadrilateral mesh - Produce (n-1)×(m-1) quadrilateral for n×m array of vertices.
- 30. Quadric Surfaces Describedwith second degree equations Quadric surfaces include: Spheres Ellipsoids Torus
- 31. Sphere A sphericalsurface with radius r centred on the origin is defined as the set of points (x, y, z) that satisfy the equation This can also be done in parametric form using latitude and longitude angles 2222 rzyx sin sincos coscos rz ry rx 22 y axis z axis x axis P ( x, y, z ) θ φ r
- 32. Ellipsoid An extensionof a spherical surface Where the radii in three mutually perpendicular directions ,have different values. parametric form using latitude and longitude angles 1 222 zyx r z r y r x sin sincos coscos z y x rz ry rx 22
- 33. Torus Doughnut –shapedobject. parametric form using latitude and longitude angles 1 2 2 22 zyx r z r y r x r sin sin)cos( cos)cos( z y x rz rry rrx
- 34. SuperQuadrics A generalizationof quadric surfaces, formed by including additional parameters into quadric equations Increased flexibility for adjusting object shapes. Superellipse 1 /2/2 S y S x r y r x
- 35. When s=1,get an ordinary ellipse Parametric representation. s y s x ry rx sin cos
- 36. Superellipsoid For s1=s2=1,get an ordinary ellipsoid Parametric representation. 1 1 12 21 /2 //2/2 S z SSS y S x r z r y r x 1 21 21 sin sincos coscos s z ss y ss x rz ry rx 22
- 38. Spline Representation Splineis a flexible strip used to produce a smooth curve through a designed set of points. Spline mathematically describe with a piecewise cubic polynomial function whose first and second derivative are continuous across the various curve section.
- 39. A splinecurve is specified using a set of coordinate position called control points , which indicates the general shape of the curve. There are two ways to fit a curve to these points: Interpolation - the curve passes through all of the control points. Approximation - the curve does not pass through all of the control points, that are fitted to the general control-point path
- 40. The splinecurve is defined, modified and manipulated with operation on the control points. The boundary formed by the set of control points for a spline is known as a convex hull
- 41. A polylineconnecting the control points is known as a control graph. Usually displayed to help designers keep track of their splines.
- 42. Parametric Continuity Condition For the smooth transition from one section of a piecewise parametric curve to the next, impose continuity condition at the connection points. Each section of a spline is described with parametric coordinate functions )(zz )( )( 21 u uuuuyy uxx
- 43. Zero –order Parametriccontinuity (C0 ) Simply means that the curves meet. That is x,y and z evaluated at u2 for the first curve section are equal to the values of x,y and z evaluated at u1 for the next curve section. First –order Parametric continuity (C1 ) The first parametric derivatives(tangent lines) of the coordinate functions for two successive curve sections are equal at their joining point. Second –order Parametric continuity (C2 ) Both first and second parametric derivatives of the two curve sections are the same at the intersection.
- 44. Geometric Continuity Condition In Geometric Continuity ,only require parametric derivatives of the two sections to be proportional to each other at their common boundary Zero –order Geometric continuity (G0 ) Same as Zero –order parametric continuity. That is the two curves sections must have the same coordinate position at the boundary point. First –order Geometric continuity (G1 ) The first parametric derivatives(tangent lines) of the coordinate functions for two successive curve sections are proportional at their joining point. Second –order Geometric continuity (G2 ) Both first and second parametric derivatives of the two curve sections are proportional at their boundary.
- 45. Spline Specification Threemethods for specifying a spline representation. 1. We can state the set of boundary conditions that are imposed on the spline. 2. We can state the matrix that characterizes the spline. 3. We can state the set of blending functions.
- 46. Parametric cubicpolynomial representation for the x coordinate of a spline section Boundary condition set on the endpoint coordinates x(0) and x(1) and on first parametric derivatives at the endpoints x’(0) and x’(1). 10,)( 23 uducubuaux xxxx
- 47. From theboundary condition, obtain the matrix that characterizes the spline curve. geomspline geomspline MMUux MMC CU x d x c x b x a uuuux )( 123)(
- 48. Finally the polynomialrepresentation 3 0 )()( k kk uBFgux
- 49. Bezier Curve andSurfaces This spline approximation method developed by the French engineer Pierre Bezier for use in the design of Renault automobile bodies. Easy to implement. Available in CAD system, graphic package, drawing and painting packages. Bezier Curve A Bezier curve can be fitted to any number of control points. Given n+1 control points position pk=(xk, yk, zk) 0≤k≤n
- 50. The coordinatepositions are blended to produce the position vector P(u) which describes the path of the Bezier polynomial function between p0 and pn The Bezier blending functions BEZk,n(u) are the Bernstein polynomials n k nkk uuBEZpuP 0 , 10),()( knk nk uuknCuBEZ )1(),()(,
- 51. where parameters C(n,k)are the binomial coefficients The individual curve coordinates can be given as follows )!(! ! ),( knk n knC n k nkk uBEZxux 0 , )()( n k nkk uBEZzuz 0 , )()( n k nkk uBEZyuy 0 , )()(
- 52. Properties Of BezierCurves Bezier Curve is a polynomial of degree one less than the number of control points
- 53. Bezier Curvesalways passes through the first and last control points. P(0) = p0 P(1) = pn Bezier curves are tangent to their first and last edges of control garph. The curve lies within the convex hull as the Bezier blending functions are all positive and sum to 1 1)( 0 , n k nk uBEZ
- 54. Design Techniques ClosedBezier curves are generated by specifying the first and last control points at same position. Specifying multiple control points at a single coordinate position gives more weight to that position.
- 55. Cubic Bezier Curve Cubic Bezier curves are generated with 4 control points. Cubic Bezier curves gives reasonable design flexibility while avoiding the increased calculations needed with higher order polynomials. The blending functions when n = 3 3 3,3 2 3,2 2 3,1 3 3,0 )1(3 )1(3 )1( uBEZ uuBEZ uuBEZ uBEZ
- 56. At u=0,BEZ0,3=1, and at u=1, BEZ3,3=1. thus, the curve will always pass through control points P0 and P3. The functions BEZ1,3 and BEZ2,3, influence the shape of the curve at intermediate values of parameter u. The resulting curve tends toward points P1 and P3.
- 57. Bezier Surface Twosets of orthogonal Bezier curves are used to design surface. Pj,k specify the location of the control points. n k nkmjkj m j uBEZvBEZpvuP 0 ,,, 0 )()(),(
- 58. B-Spline Curves and Surfaces 1.The degree of a B-spline polynomial can be set independently of the number of control points. 2. B-splines allow local control over the shape of a spline curve (or surface)
- 59. The pointon the curve that corresponds to a knot is referred to as a knot vector. The knot vector divide a B-spline curve into curve subinterval, each of which is defined on a knot span.
- 60. Given n+ 1 control points P0, P1, ..., Pn Knot vector U = { u0, u1, ..., un+d } The B-spline curve defined by these control points and knot vector Pk is kth control point Blending function Bk,d of degree d-1 n k nduuu dk B k puP 0 12, maxmin ,)( , )(
- 61. Blending functionsdefined with Cox-deBoox recursive form )()()( ,0 ,1 )( 1,1 1 1, 1 , 1 1, uB uu uu uB u uu uB otherwise uuif uB dk kdk dk dk dk k dk kk k
- 62. To change theshape of a B-spline curve, modify one or more of these control parameters: 1. The positions of control points 2. The positions of knots 3. The degree of the curve
- 63. Uniform B-Spline Thespacing between knot values is constant. Non-uniform B-spline Unequal spacing between the knot values. Open uniform B-Spline This B-Spline is across between Uniform B-Spline and non-uniform B-Spline. The knot spacing is uniform expect at the ends where knot values are repeated d times
- 64. B-Spline Surfaces Similar toBezier surface 2 2 221121 1 1 0 ,,, 0 )()(),( n k dkdkkk n k vBuBpvuP
- 65. Sweep Representations Sweeprepresentations are useful for constructing 3 dimensional objects that possess translational, rotational or other symmetries. Objects are specified as a 2 dimensional shape and a sweep that moves that shape through a region of space
- 67. Octrees Octrees arehierarchical tree structures used to represent solid objects. Octrees are particularly useful in applications that require cross sectional views – for example medical applications.
- 68. Octrees & Quadtrees Octrees are based on a two-dimensional representation scheme called quadtree encoding. Quadtree encoding divides a square region of space into four equal areas until homogeneous regions are found. These regions can then be arranged in a tree
- 69.
- 70. An octreetakes the same approach as quadtrees, but divides a cube region of 3D space into octants. Each region within an octree is referred to as a volume element or voxel. Division is continued until homogeneous regions are discovered