Combinations to Form Geometry Models

Geometric modelling is the mathematical representation of the geometry of an object. Curves are used to develop these models. The different types of geometric modelling are as follows:

a) Solid modelling: Another name for solid modelling is volume modelling. The given object is defined in terms of its edges, nodes and surfaces. It gives a perfect and precise mathematical representation of the volume filled within the object. It is based on the rules of topology that all the surfaces of the object are sewn together accurately. The process of solid modelling is built on the “half-space” concept. Solid modelling is used in weight and volume calculations, code generation, robotic simulation and much more.

b) Surface modelling: It represents the solid appearing object. It is a complicated method of modelling, such as a car, ship or aeroplane body, but not as improved as solid modelling. The surface and solid models look similar, the former cannot be cut open the way solid models can be. Surface modelling is used in situations where changes are needed for one or more faces, as applying changes to this type of geometry is hard. It allows building one face at once, such that there is control over the exact shape and direction of that particular face. Surfaces are mostly useful as a geometry reference in a step towards building the desired face of the model.

c) Wireframe modelling: It is one of the oldest modelling techniques for denoting a solid model. It involves lines and vertices that are usually done in designing a 3D real-world object. It was discovered in the 1960s. It is also referred to as an edge representation or stick figure. The lines present inside the wireframe are used to connect to develop polygons, such as triangles and rectangles, when they are fixed together. Wireframe modelling is used in comparing a three-dimensional model to its source, demonstrating concepts in an easy and quick way. A skeletal outline can be made without presenting the in-depth details, as it is simple and is detectable to others.

Different examples where a combination is used in geometric modelling are as follows:

1) If there exist n points in a plane that are not collinear, then by joining them, we can obtain ⁿC₃ triangles and ⁿC₂ straight lines.

2) If there exist n points in a plane out of which m are collinear, then by joining them, we get ⁿC₂ – ^mC₂ + 1 straight line and ⁿC₃ – ^mC₃ triangles.

3) The number of diagonals of a polygon with n sides and with n vertices is ⁿC₂ – n = number of diagonals = [n (n – 3)] / 2, where n is the number of sides.

4) If a set of m parallel lines are intersected by another set of n parallel lines, then the number of parallelograms we can obtain is ^mC₂ x ⁿC₂.

5) The maximum number of points of intersection of

a) n straight lines in a plane is ⁿC₂.

b) n unequal circles in a plane is ⁿC₂ x 2.

Example 1: There are 12 collinear points in a place, of which 7 are collinear. By joining them, we can have

a) number of straight lines = ¹²C₂ – ⁷C₂ + 1

b) number of triangles = ¹²C₃ – ⁷C₃

c) number of circles = ¹²C₃ – ⁷C₃

d) number of pentagons = ¹²C₅ – ⁷C₅ – (⁷C₄ x ⁵C₁) – (⁷C₃ x ⁵C₂)

Combinations to Form Geometry Models – Video Lesson