Constructive Methods in Modelling Lecture 7 (Modelling)

Surfaces of Revolution lCreates circular symmetric objects. lCreate a 3D surface by revolving a 2D profile curve around an axis of rotation in space. lClosed profile curves generate closed surfaces. lExamples: 1.Circular cylinder: The profile is a line segment parallel but not coincident with the axis of rotation. The closed version requires a rectangle. 2.Truncated cone: The line segment profile is slanted with respect to the axis.

Examples: Surfaces of Revolution 3.Torus: The profile is a circle inset in a plane aligned with the axis. 4.Complex circularly symmetric shapes: employ a Bézier or B- spline profile curve.

Generalized Cylinders lExtrusion: sweep a 2D shape along a (non-circular) path. lSome objects can be generated by extrusion or revolution. e.g. a cylinder (an extruded circle or a revolved line). lGeneralized cylinders extend the concept of extrusions and surfaces of revolution to the extreme. lTotal control over all sweep parameters. lBut can produce degeneracies, e.g. self-intersection.

Generalized Cylinder Parameters lCan vary: 1.Cross Section – 2D shape does not have to be a circle and can change shape as it is swept. 2.Sweep Path – path does not have to be a straight line or revolution, can be any space curve. 3.Twist – the cross section can be rotated as it moves along the path 4.Scale – the size of the cross section can change along the path 5.Normal vector direction – conventionally the vector normal to the cross section points along the path, but even this can be varied 6.And any other parameters.

Spatial Deformation lPrinciple:  Indirectly deform an object by warping the surrounding space. lJelly metaphor:  A shape is set within a block of jelly.  Flexing the jelly results in a corres- ponding distortion of the shape. lMechanism:  Object vertices are embedded in a parametric hyperpatch, which is a 3D generalization of B-spline curves.

Free Form Deformation lHyperpatch Equation:  Vertices (q) are expressed as a sum of control points (p) weighted by B-spline basis functions (b). lFFD Algorithm: [1] Establish a (u,v,w) hyperpatch parameterization for all object vertices. [2] Displace control points. [3] Apply the hyperpatch equation.

Illustration of FFD Pre-Deformation Post-Deformation

Evaluation of FFD lAdvantages: Fast Smooth, sculpted results Variable scope lDisadvantages:  Lack of precise control  Screen clutter  Counter-intuitive lLater Developments: 1.Direct Manipulation with points or curves 2.Different deformation boundaries.

Constructive Solid Geometry lConstructive Solid Geometry (CSG) consists of regularized boolean set operations on closed 3D objects. lBoolean Set Operations Union Intersection Difference Sphere (A) and Parallelepiped (B)

CSG: Robustness lRegularized (represented by *): closed input objects produce a closed output object. lCSG arguments may be analytic (RT) or polygon mesh (PSC) objects (or both). lCSG is prone to special cases which cause robustness problems.  If points, edges, or faces of two polygon mesh arguments coincide an incorrect object may result.  For example, in modelling a gaming die, if the top of the cylindrical pips are flush with the cube surface then the difference may not be visible. lCSG is a huge area of research. Entire books have been written on this topic.

CSG Tree lCSG is evaluated as a pre-process in PSC and at run-time for RT. lThe CSG tree is an effective structure for evaluating ray-object intersections. lLeaf nodes are objects. lInternal nodes are boolean operations.

CSG: Ray Tracing Algorithm Select an eye point and a screen plane FOR every pixel in the screen plane [num. pixels depends on image resolution] FOR every leaf node in the CSG tree compute and order by increasing depth the ray-object intersections. ENDFOR recurse left subtree returning intersection list [A] recurse right subtree returning intersection list [B] interleave sort A and B by increasing value along ray. FOR all entries in intersection list keep track of current state and use current intersection to access lookup table. ENDFOR RETURN new intersection list. ENDFOR A~AB~B ABCout A~BCin ~AB ~A~B A Intersection B

CSG: Ray Tracing Example

Constructive Modelling Exercise lSpecify the steps required for construction of the jug pictured below. Note: some stages can be achieved in several different ways. Enumerate them where possible.

Constructive Modelling Solution 1.Jug: Surface of Revolution 2.Handle: Extrude Ellipse 3.Spout: Free-Form Deformation 4.Combination: Union of Deformed Jug and Handle