An alternative to parametric method is the use of implicit representation where the surface is defined as the zero contour of a function of 2 or 3. Several models have been developed for representing blobby objects as distribution In computer Graphics, a spline surface can be described with two sets of. Several models have been developed for representing blobby objects as In computer graphics, the term spline curve now refer to any composite curve formed.
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Published by Frank Golden Modified over 3 years ago. Build mesh by hand: Tessellate a theoretical smooth surface: Tessellation is the process of creating a polygonal approximation from a smooth surface. Extrusion is the process of moving a 2D cross-section through space to create a 3D solid.
Revolution is the process of rotating a 2D cross-section about an axis to create a 3D solid. Creations – Create boobby geometry from some other mathematical object. Binary Creations – Create a new mesh from a binary operation of two other meshes. Deformations – Move only the vertices of a mesh. Manipulations – Modify the geometry of the mesh, but not necessarily topology. Measurements – Compute some value of the mesh. The decimation algorithm, designed to reduce iso-surfaces containing millions of polygons, is quite fast.
Problems with polygonal models: They approximate smoothly curving surfaces. Tradeoff between realism and efficiency. Sphere, ellipsoid, torus and cone come under general quadrics. Quadratic surfaces, particularly spheres and ellipsoids are common elements of graphic compputer.
Sphere Ellipsoid Torus Cone 7. Increased flexibility for adjusting object shapes. Super Ellipse Ellipse equation can represent super ellipse by allowing the exponent on the x and y terms to be variable. Super Ellipsoid Ellipsoid equation can represent super ellipsoid by incorporating two exponent parameters. Super ellipses plotted with diff bpobby for parameters s and with radius equal compyter both the axes.
Super ellipsoids plotted with diff values for parameters s1 and s2, radius on all the axes are equal 9. These objects change their molecular shapes very easily, so they cannot be described simply with spheres or elliptical shapes.
Need to model surface shapes so that the total volume remains constant. Several models have been developed to handle these kind of objects.
This property allows us to design complex shapes with lower degree. If the curve is in a plane resp. The affine invariance property — Applying geometric or even affine transformation to computfr B-spline graphjcs, then the transformation can be applied to control points. Therefore, we do not have to transform the curve. Control points influence the directions the surface takes.
CSG is a modeling technique that uses Boolean operations to combine 3D solids. The Boolean operators used by CSG are: Union, Intersection, Difference or Minus. An object is stored as a tree with operators at the internal nodes and simple primitives at the leaves. Used in almost all engineering CAD packages. Used in ray tracing and particle transport. Used for some manufacturing or engineering computation applications.
Few generic modelling languages and software are also supported by CSG. The picture area is divided into 4 sections. Those 4 sections are then further divided into 4 subsections. Continue blohby process, repeatedly dividing a square region by 4. Then impose a limit to the levels of division otherwise we could go on dividing the picture kbjects. First three levels of a quad tree. Recursive pictures like in the figure can easily implemented using quadtrees.
The octree is constructed by recursively subdividing each cube into eight objevts, starting at the root node a single large cube. Each cube in an octree can be one of three colors. Inn black cubes and white cubes are leaf nodes in the tree. A gray cube lies on the boundary of the object and is only partially filled. It is an interior node of the tree and has eight equally sized children of different colors.
What tends to be difficult is managing the geometry needed blobbh divide the cubes and store the details at each node. As in the case of the quadtree the octree is often built up dynamically as data become available. In this case dynamic construction often results in an unbalanced tree with areas of space being covered more finely than others. Statistical when random parameters are involved. Amount of variation of a self similar object.
Blobby Objects |authorSTREAM
Fragmentation, roughness of the object. The fractal dimension of a self-similar fractal with a single scaling factor s is obtained using ideas from subdivision of a Euclidean object.
We can define the fractal dimension similarly with number of subparts n and a given scaling factor s. The Julia set is closed, the Fatou set is open. rgaphics
An Overview of Metaballs/Blobby Objects
The Julia and Fatou sets are disjoint. Connected components of this set are called the Fatou components of P. Immediate basin of attraction attracting petal Siegel disc Herman ring Baker domain. The image is scanned and converted to grayscale and the natural log of black points are plotted which helps in finding the fractal dimension of the leaf.
In order to determine the perimeter of a leaf, the outline of the leaf needed to be identified. After the image was converted into blobvy binary image, an edge detector and boundary will be used to create an image that is completely black with a pixel wide white outline of the leaf.
Midpoint displacement takes a straight line and turns it into a ragged line that can look like the outline of a mountain range. The resulting fractal has a dimension between one and two.
Start with a straight line between points A and B. Create the midpoint, M, half way between A and B. Then displace the midpoint up or down by a random value. This creates two new lines joined at the midpoint.
Repeat this displacement algorithm recursively for each new line segment. At each step, we also adjust the amplitude of the displacement by some method. One usually reduces it by an exponential function.
However it is fairly homogenous and isotropic. It tends to look fake to the eye. Midpoint displacement can be done in higher dimensions as well, but the process gets more and more complex. For 2D terrain, which is useful for RL terrain generation, we need to displace the midpoint of a square and partition the square into four smaller ones, which are then recursively displaced.
The algorithm is know as Diamond-Square, referring to the two steps of displacement then partitioning. An L-system involves three main components: Then, z is repeatedly updated using: After numerous iterations, if the magnitude of z is less than 2 we say that pixel is in the Julia set and color it accordingly.
Performing this calculation for a whole grid of pixels gives a fractal image. For each transformation, the shape is duplicated into each half of the resulting shape, so it gets twice the amount of detail or number of lumps in the shape. For Julia sets, c is the same complex number for all pixels, and there are many different Julia sets based on different values of c. By smoothly changing c we can transform from one Julia set to another over time, creating animated fractal shapes. The Mandelbrot set can be considered a map of all Julia sets because it uses a different c at each location, as if transforming from one Julia set to another across space.
A specific Julia set can be defined by a point in the Mandelbrot set matching its constant c value, and the look of an entire Julia set is usually similar in style to the Mandelbrot set at that corresponding location. Points near the edges of the Mandelbrot set typically give the most interesting Julia sets. To realize the 3D fractal, the quaternion as the iteration variable is applied to create the 4D Julia set. Quaternion Julia fractals are created by the same principle as the more traditional Julia set except that it uses 4 dimensional complex numbers instead of 2 dimensional complex numbers.
The approach used here is to intersect the 4D solid with a plane, in essence this makes one of the quaternion components dependent on the other three. To get a feel for the true nature of the quaternion fractal one needs create a whole series of slices along an axis perpendicular to the slice plane. The following shows 6 slices moving the cutting plane along one of the quaternion axes from the origin in steps of 0. A shape grammar consists of shape rules and a generation engine that selects and processes rules.
A shape grammar minimally consists of three shape rules: