Identifying the Critical Points of Skeleton-Based Convolution Surfaces for Conceptual Design

2007 ◽  
Author(s):  
Guohua Ma ◽  
Richard H. Crawford
Keyword(s):  
Critical Points ◽  
Three Dimensional ◽  
Analytic Solutions ◽  
Field Function ◽  
Solid Models ◽  
Convolution Surfaces ◽  
Will Force ◽  
Dimensional Object ◽  
Skeletal Model ◽  
Parameter Values

Skeletal modeling is an approach to creating solid models in which the engineer designs with lower dimensional primitives such as points, lines, and triangles. The skeleton is then “skinned over” to create the surfaces of the three dimensional object. Convolution surfaces are generated by convolving a kernel function with a geometric field function to create an implicit surface. Certain properties of convolution surfaces make them attractive for skeletal modeling, including: (1) providing analytic solutions for various geometry primitives (including points, line segments, and triangles); (2) generating smooth surfaces (3) and providing well-behaved blending. We assume that engineering designers expect the topology of a skeletal model to be identical to that of the underlying skeleton. However the topology of convolution surfaces can change arbitrarily, making it difficult to predict the topology of the generated surface from knowledge of the topology of the skeleton. To address this issue, we apply Morse theory to analyze the topology of convolution surfaces by detecting the critical points of the surface. We describe an efficient algorithm that we have developed to find the critical points by analyzing the skeleton. The intent is to couple this algorithm with appropriate heuristics for determining parameter values of the convolution surface that will force its topology to match that of the skeleton.

Conceptual Design by a Topology Consistent Skeletal Modeler With Convolution Surfaces

2013 ◽  
Author(s):  
Guohua Ma ◽  
Richard H. Crawford
Keyword(s):  
Conceptual Design ◽  
Three Dimensional ◽  
Design Concept ◽  
Design Stage ◽  
Field Function ◽  
Design Engineers ◽  
Convolution Surfaces ◽  
Dimensional Object ◽  
Skeletal Model ◽  
Lower Dimensional

During the conceptual design stage, the design engineers usually sketch their design ideas. For those sketches, the skeleton of the design idea can be created with lower dimensional primitives like lines, arcs, etc. In this paper, we focus on skeletal modeling, which is an approach to creating solid models in which the engineer designs with lower dimensional primitives such as points, lines, and triangles. The skeleton is then “skinned over” to create the surfaces of the three dimensional object. Then the convolution surfaces are generated by convolving a kernel function with a geometric field function to create an implicit surface. We propose that skeleton, even it is simple, contains important design information, such as the geometric, topology that defines the design concept. It is very important to keep the topology of the skeleton and thus the important information that defines the design concept, i.e, the geometry of the product, the functionality of the product determined by the topology of the design. We assume that design engineers expect the topology of a skeletal model to be identical to that of the underlying skeleton. In this paper, the system is described and some examples are illustrated to use the skeletal based modeler.

Reconstruction of Objects from their Projections Using Orthogonal Functions

1973 ◽  
Vol 31 ◽  
pp. 268-269
Author(s):  
Elrnar Zeitler
Keyword(s):  
Unit Area ◽  
Three Dimensional ◽  
One Dimension ◽  
Spatial Density ◽  
Dimensional Representation ◽  
Orthogonal Functions ◽  
Object A ◽  
Plane Normal ◽  
Dimensional Object ◽  
Spatial Density Distribution

Considering any finite three-dimensional object, a “projection” is here defined as a two-dimensional representation of the object's mass per unit area on a plane normal to a given projection axis, here taken as they-axis. Since the object can be seen as being built from parallel, thin slices, the relation between object structure and its projection can be reduced by one dimension. It is assumed that an electron microscope equipped with a tilting stage records the projectionWhere the object has a spatial density distribution p(r,ϕ) within a limiting radius taken to be unity, and the stage is tilted by an angle 9 with respect to the x-axis of the recording plane.

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