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Chapter 10: Problem 13

Each collection that follows represents the vertices (using the traditional rectangular coordinate system) of a patch in a polygonal mesh. Describe the shape of the mesh. Patch 1: \((0,0,0)(0,4,0)(2,4,0)(2,0,0)\) Patch 2: \((0,0,0)(0,4,0)(1,4,1)\) \((1,0,1)\) Patch 3: \((2,0,0)(1,0,1)(1,4,1)\) \((2,4,0)\) Patch 4: \((0,0,0)(1,0,1)(2,0,0)\) Patch 5: \((2,4,0)(1,4,1)(0,4,0)\)

Short Answer

Expert verified
The overall shape is a three-dimensional trapezoidal prism or frustum-like shape.

Step by step solution

01

Identify shapes of individual patches

Start by identifying the shape of each patch based on the given vertices. Each patch consists of either 3 or 4 vertices. A patch with 4 vertices is generally a quadrilateral, while a patch with 3 vertices is a triangle.
02

Describe Patch 1

Patch 1 has vertices \(0,0,0\), \(0,4,0\), \(2,4,0\), \(2,0,0\). These lie on the \(x-y\) plane with \(z=0\) and form a rectangle.
03

Describe Patch 2

Patch 2 has vertices \(0,0,0\), \(0,4,0\), \(1,4,1\), \(1,0,1\). These form a quadrilateral where the heights differ, meaning it is a slanted rectangle or a trapezoidal shape, sloping upwards along the \(z\) axis.
04

Describe Patch 3

Patch 3 has vertices \(2,0,0\), \(1,0,1\), \(1,4,1\), \(2,4,0\). This patch forms a trapezoid, similar to Patch 2, but on the side from \(x = 2\) to \(x = 1\), sloping along the \(z\) axis.
05

Describe Patch 4

Patch 4 has vertices \(0,0,0\), \(1,0,1\), \(2,0,0\). This forms a triangle on the base plane that slopes upward to the point \(1,0,1\), a triangle with a vertical raise.
06

Describe Patch 5

Patch 5 has vertices \(2,4,0\), \(1,4,1\), \(0,4,0\). This forms a triangle on the top plane, connecting the vertices at \(z = 0\) and \(z = 1\).
07

Synthesize Overall Shape

Combine the descriptions of all patches to derive the overall shape of the mesh. Patch 1 forms the base, while Patches 2 and 3 form slanted sides. Patches 4 and 5 close the shape with triangular sides, forming a frustum-like shape (trapezoidal prism).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate System
A coordinate system is a way to specify the position of points in space. When we talk about points in geometry, we often refer to coordinates, which are specific sequences of numbers representing a location. Common types of coordinate systems include:
  • Rectangular (Cartesian) Coordinate System: Used to determine the positions in two or three-dimensional spaces using perpendicular axes, usually labeled as X, Y, and Z.
  • Polar Coordinate System: Used for two-dimensional planes where each point is determined by a distance from a reference point and an angle from a reference direction.
  • Spherical and Cylindrical Coordinate Systems: Extend the idea of polar coordinates to three-dimensional spaces.
In the original exercise, the traditional rectangular coordinate system is used. Each vertex of the polygonal patches is defined by three coordinates (x, y, z), indicating their positions in a 3D space.
Vertices
In geometry, vertices are the corner points of a shape. Each vertex is defined by its coordinates in a given coordinate system. For example, in a 3D rectangular coordinate system, a vertex is defined by an ordered triplet of numbers (x, y, z). Vertices are crucial in forming and defining the shape of polygonal meshes:
  • In Patch 1, vertices at coordinates such as (0,0,0) and (2,4,0) create the boundaries of the rectangle on the x-y plane.
  • Patches 2 and 3 use vertices like (1,0,1) and (1,4,1) to create sloped planes by varying the z-coordinate.
By connecting vertices with edges, you form the faces of the polygonal shape. The unique combination and manipulation of the coordinates determine the overall geometry of the shape.
Trapezoidal Prism
A trapezoidal prism is a type of polyhedron. It features two parallel faces called bases that are trapezoids, and other faces that are typically rectangles or parallelograms, forming a prism with trapezoidal bases.
  • Base Composition: The base of a trapezoidal prism is a trapezoid, which is a 4-sided figure with at least one pair of parallel sides.
  • Height: The distance between the two bases of a trapezoidal prism.
In the context of the original exercise, the mesh forms a trapezoidal prism structure. Patches like Patch 1 instantiate a rectangular base, while Patches 2 and 3 consist of slanted trapezoidal faces that connect different heights in the z-dimension, giving the prism its trapezoidal shape. Patches 4 and 5 close the sides with triangular faces.
Rectangular Coordinate System
The rectangular coordinate system, also known as the Cartesian coordinate system, uses three perpendicular axes (commonly X, Y, and Z) to describe the position of points in space. Each point is represented by coordinates (x, y, z):
  • X-Axis: Represents the horizontal dimension.
  • Y-Axis: Stands for the vertical dimension in a 2D plane, or the second horizontal dimension in a 3D space.
  • Z-Axis: Is the third dimension, showing height or depth in 3D space.
In the exercise, these axes help locate the vertices of each patch. For instance, (2, 4, 0) indicates 2 units along the x-axis, 4 units along the y-axis, and stays on the plane at z = 0. Understanding this system is key to visualizing the geometric shapes and configurations that result in complex structures like the trapezoidal prism in polygonal meshes.

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Most popular questions from this chapter

When using a perspective projection, under what conditions will a sphere in the scene not produce a circle on the projection plane?

Suppose the points \((1,0,0),(1,1,1)\), and \((1,0,2)\) are the vertices of a planar patch. Which of the following line segments is/ are normal to the surface of the patch? a. The line segment from \((1,0,0)\) to \((1,1,0)\) b. The line segment from \((1,1,1)\) to \((2,1,1)\) c. The line segment from \((1,0,2)\) to \((0,0,2)\) d. The line segment from \((1,0,0)\) to \((1,1,1)\)

What is the distinction between anisotropic surfaces and isotropic surfaces?

Suppose the points \((1,0,0),(1,1,1)\), and \((1,0,2)\) are the vertices of a planar patch and the vertices are listed in a counter- clockwise order as seen from outside the object. In each case that follows, indicate whether a ray of light originating at the given point would strike the surface of the patch from outside or inside the object. a. \((0,0,0)\) b. \((2,0,0)\) c. \((2,1,1)\) d. \((3,2,1)\)

Suppose one end of an eight-foot straight pole is four feet from the center of projection. Moreover, suppose that a straight line from the center of projection to one end of the pole intersects the projection plane at a point that is one foot from the center of projection. If the pole is parallel to the projection plane, how long is the image of the pole in the projection plane?
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