Implicit 3D Shape Manipulation Using Coparametrization and Jacobians
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Solution Overview
Problem
Existing methods for directly manipulating composite 3D shapes assembled from implicitly defined 3D primitives are limited, as they lack metadata or additional information necessary for tracking surface points and establishing a connection between geometrical updates and parameter changes, making direct manipulation difficult and time-consuming.
Innovation Solution
A computing system uses coparametrization and Jacobian matrices to determine parameter changes based on user inputs, enabling direct manipulation of implicitly defined 3D shapes by converting point representations into coparametrizations and calculating transformations using Jacobian operators to update parameters in real-time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If procedural methods with implicit definitions are used to generate 3D shapes, then the shapes can be compactly represented and efficiently generated, but direct manipulation becomes difficult and time-consuming due to lack of metadata for tracking surface points
Solution Approach 1:
The patent introduces coparametrization values as an intermediary representation that bridges the implicit mathematical definition and the visual surface points. This intermediary enables tracking of surface points without changing the compact implicit representation, allowing direct manipulation while maintaining generation efficiency.
Solution Approach 2:
The patent transforms geometric operations on surface points into parameter changes by computing Jacobian matrices that relate coparametrization space to parameter space. This allows direct manipulation of the shape through intuitive geometric operations while automatically updating the underlying parameters.
2Ease of operation
If sliders are used to adjust procedural parameters, then parameter values can be manually controlled, but obtaining desired graphics requires experimentation and is time-consuming
Solution Approach 1:
The system provides immediate visual feedback when users interact with surface points, automatically computing the corresponding parameter changes and updating the shape in real-time. This eliminates the trial-and-error process by directly translating user intent into parameter adjustments with instant visualization of results.
Solution Approach 2:
The patent replaces the mechanical slider-based parameter adjustment system with a direct geometric manipulation system. Instead of manually moving sliders to change parameters, users directly interact with surface points and the system automatically computes and applies the necessary parameter changes through Jacobian-based transformations.
3Use of energy by moving object
If implicit definitions without metadata are used, then computational resources are saved, but establishing connection between geometrical updates and parameter changes becomes difficult
Solution Approach 1:
The patent segments the information representation into two distinct parts: the compact implicit mathematical definition and the coparametrization metadata. This segmentation allows the system to maintain computational efficiency from the implicit definition while adding minimal metadata overhead for tracking surface points and enabling direct manipulation.
Data Source
AI summary
Techniques are disclosed for direct manipulation of implicitly defined digital three-dimensional (3D) shapes. In an example method, a computing device renders a 3D shape based on an implicit definition including one or more parameters. The computing device receives an indication of an input indicating a modification to the 3D shape at a point. The computing device determines an alternative representation of the point. The computing device determines a position of the point based on the alternative representation. The computing device determines a transformation that relates the position to the one or more parameters. The computing device determines a change in at least one parameter based on the transformation and the input. The computing device re-renders the 3D shape based on the implicit definition and the change in the at least one parameter. The re-rendered 3D shape includes the modification indicated by the input.