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Math

2D Shape Representation

Silvia Sellán’s talks on Day 3 of SGI gave a large-scale overview of shape representation, which I really appreciated as a Geometry Processing newbie. 

First, what is Shape Representation? To describe Geometry Processing, Silvia quoted Alec Jacobson: “Geometry Processing is a subfield of biology.” Just as biological organisms have a life cycle of birth–growth–death, so do shapes: they are born (created in a computer program), then they are manipulated and changed through various algorithms, and then they die–or rather, they leave the computer and go into the world as a finished product, such as a rendered image in a video game. Shape Representation encompasses the methods by which a shape can be created. 

Silvia gave us an introduction to Shape Representation by focusing first on 2D methods. Using the example of a simple curve, we looked at four ways of representing the curve in 2D. Each method has its advantages and disadvantages:

The simplest way of representing a curve is a point cloud, or a finite subset of points on the curve. The advantage is that this information is easy to store, since it’s just a list of coordinates. However, it lacks information about how those points are connected, and one point cloud could represent many different shapes.

The second method is a polyline or polygonal line, which consists of a point cloud plus a piecewise linear interpolation between connected points. Like a point cloud, a polyline is easy to store. It’s also easy to determine intersections between them, and easy to query–to determine whether any given point lies on the polyline. However, since a polyline is not differentiable at the vertices, it does not allow for differential quantities such as tangents, normal vectors, or curvature. It also needs many points to be stored in order to visually approximate the desired curve. 

Third, we can use a spline, which is a piecewise polynomial interpolation with forced continuous derivatives. Splines have differential continuity and don’t need many points in order to approximate a curve but are more difficult to query and to solve intersections. 

Finally, we have implicit shape representations, which use implicit functions to define a shape as the region of the plane where the function equals zero. The advantages of implicit representations include easy computation of boolean operations, and it is also possible to treat the shape as an image and use image processing tools and machine learning algorithms. However, not every shape can be represented implicitly, and they are hard to manipulate for many graphics applications. 

These methods of shape representation in 2D formed the scaffolding for us to later build an understanding of shape representation in 3D.

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