1 / 16

Constructive Methods in Modelling

Constructive Methods in Modelling. Lecture 7 (Modelling). Surfaces of Revolution. Creates circular symmetric objects. Create a 3D surface by revolving a 2D profile curve around an axis of rotation in space. Closed profile curves generate closed surfaces. Examples:

lance-bruce
Download Presentation

Constructive Methods in Modelling

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Constructive Methods in Modelling Lecture 7 (Modelling)

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

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

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

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

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

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

  8. Illustration of FFD Pre-Deformation Post-Deformation

  9. Evaluation of FFD • Advantages: • Fast • Smooth, sculpted results • Variable scope • Disadvantages: • Lack of precise control • Screen clutter • Counter-intuitive • Later Developments: • Direct Manipulation with points or curves • Different deformation boundaries.

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

  11. CSG: Robustness • Regularized (represented by *): closed input objects produce a closed output object. • CSG arguments may be analytic (RT) or polygon mesh (PSC) objects (or both). • CSG 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. • CSG is a huge area of research. Entire books have been written on this topic.

  12. CSG Tree • CSG is evaluated as a pre-process in PSC and at run-time for RT. • The CSG tree is an effective structure for evaluating ray-object intersections. • Leaf nodes are objects. • Internal nodes are boolean operations.

  13. 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 Intersection B

  14. CSG: Ray Tracing Example

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

  16. Constructive Modelling Solution • Jug: Surface of Revolution • Handle: Extrude Ellipse • Spout: Free-Form Deformation • Combination: Union of Deformed Jug and Handle

More Related