2somvan

= = // *~*6.1-- Solid Shapes*~* //

__Isometric Drawing__- Lines that represent 30° angles in a shape or object. It's a method of showing different perspectives of a figure. //**Example: **//

=

 * --The letters in the picture are isometric. When every side turns, we are shown an isometric view; we can see the different views of the letters, almost all perspectives are different from each other.**=====

__Orthographic Projection__- The different views of an object or shape. There are usually six different orthographic views of an object, including: the top, bottom, front, back, right, and left.
 * [|//Click to view an Orthographic example.//]**

__Solids__- Figures with closed sides. //**Example:**//

--An example of a solid would be this eight-sided figure. It has closed sides and is 3-dimensional.
= = // *~*6.2-- Relationship in Figure Spaces*~* //

__Polyhedron__- A figure that contains polygons and is a closed figure. Those polygons are called the faces of the figure. The edges are on the sides of the figure and connect with each other. While each corner of the figure are called vertices.
 * //Example://**

--All five of these shapes are polyhedrons because they all have the characteristics of a polyhedron. The shapes have faces and connect with their corresponding sides.
__Parallel Planes__- Planes that never intersect to any points on either plane.

__Line Perpendicular to Plane__- When a line and point P is perpendicular, and all the lines in that plane passes through point P.

__Dihedral Angle__- Two half-planes with a common edge that form an angle. The two planes are called the face and the common edge is called the edge.
 * [|Dihedral Angle within Octants]** Site

__Dihedral Angle Measurement__- Two rays on the plane's faces, which are perpendicular to the edge of the angle.

= = // *~*6.3-- 3 Dimensional Shapes called "Prisms"*~* //

__Diagonal of Polyhedron__- The vertices of two faces of the figure whose endpoints are conn

__Prism__- A figure that has a polygonal area. A figure that's image is on two parallel planes and are connected according to their corresponding edges and sides. The original images on the planes are called the bases of the prism. The leftover faces, which are quadrilaterals,(the faces that occurred from the connection of the edges) are called the lateral faces. Then the lateral edges of the prism are the edges of each lateral face of the prism not touching any of the bases.
 * //Example://

[|//Prism Site//]**

__Right Prism__- When the lateral faces are rectangles, and not any other shapes.

__Oblique Prism__- When the lateral faces have at least one non-rectangular shape. ected by a segment.
 * [|////Site with right and oblique prism////]**

__Formula for the Length of the Diagonal of a Right Rectangular Prism__- diagonal= √(length²+width²+height²). //**Example of figuring out how to find the missing dimensions:**// 1. If the length=16 in.; height=21 in.; diagonal=29 in; width=?. 29=√(16² + w² + 21²) ---> (29)²=[√(16² + w² + 21²)² ---> 841=√(697 + w²) ---> 841-697= 144=w² ---> √(144)= //**12=w**// 2. If the length=5 m.; width=10 m.; height=7 m.; diagonal=?. d=√(5² + 10² + 7²) --> d=√(25 + 100+ 49) = √(174) = **//13.2=d//** = = = = = = = = *~*// 6.4-- Three Dimensional Coordinates*~* //

__Right-Handed System__- Using the right hand, make a shape with your index, thumb, and middle finger. Extend the index and thumb finger so that they form a 90° angle. The middle finger should be bent towards you, away from the index and thumb fingers. This should create a visual dimensional picture of the x, y, and z-axis.
 * //Example://**

=
--Here's an example of a 3D graph. The x-axis is pointed towards you, the y-axis is pointed sideways, and the z-axis seems to be pointing up and down. The Right-Handed System helps visualize a graph like this one in which 3D graphing takes places.=====

__Octant__- There are eight octants that are shown on a 3D plane. The octants are described using words such as, top, bottom, front, back, right, and left. //If the x-axis point is positive, then the point is in the front octant; if the point is negative, then it is in the back octant.// If the point in the y-axis is positive, then the point is in the right octant; if the point is negative, then the point is in the left octant. //If the point in the z-axis is positive, then the point is in the top octant; if the point is negative, then the point is in the bottom octant.// //Example:// 1. //Point (4, -1, 2)// shows us that the point is in the front, left, top octant. 2. //Point (-5, 3, 8)// shows us that the point is in the back, right, top octant. 3. //Point (-6, -8, -2)// shows us that the point is in the back, left, bottom octant. And so on.

__Coordinate Planes__- There are three of these, named xy-plane, xz-plane, and yz-plane, according to where they're points are located in a dimensional plane. - Every point is 0 in the z-coordinate, if the xy-plane is used. - Every point is 0 in the y-coordinate if the xz-plane is used. - Every point is 0 in the x-coordinate if the yz-plane is used.

__Three Dimensional Distance Formula__- Distance= √[(X2-X1)² + (Y2-Y1)² + (Z2-Z1)²]. //Example:// Find the Distance between the two points. 1. //(-3, -1, -5) and (-1, -2, -3)// d=√[-1-(-3)²] + [-2-(-1)²] + [-3-(-5)²] = √(4+1+4) = √(9) = **//3 units=Distance//** 2. //(2, 1, 3) and (5, -2, 7) d=//√([(5-2)² + (-2-1)² + (7-3)²] = √(4+9+16) = √(29) = **//5.4 units=Distance//**

__Three Dimensional Midpoint Formula__- Midpoint= (X1+X2÷2), (Y1+Y2÷2), (Z1+Z2÷2).

= = // *~*6.6-- Perspectives*~* //

__Perspective Views__- The view of an image that makes a flat image seem like a 3D image. There are lines that help make the image pop out. //This site shows an example of how perspective view can be used in jobs; like creating games using perspective views.// [|//**Javascript Games**//]

__Vanishing Point__- A point located on a horizon that is connected by two parallel lines, which makes it seem as if the points intersect, but really don't. The parallel lines just aren't seen behind the vanishing point in the perspective viewed.
 * //Example://**

=
--The vanishing point on here is where the two parallel sides of the road seem like they're intersecting at the top. The three points and lines form a triangle-looking figure. The horizon shown here is the far view of the land=====

__Sets of Parallel Lines Theorem__- When two parallel lines meet at a common point(vanishing point) in a perspective drawing.

__Lines Parallel to the Ground Theorem__- A line that's not parallel to the picture plane that meets the horizon line in the drawing. Another line that meets that same horizon must be parallel to the first line in a perspective drawing.