2tuulis

=**Solid Shapes- Drawing Cubes and Orthographic Projections**= = =


 * Objectives**:
 * Use isometric dot paper to draw three- dimensional shapes composed of cubes.
 * Develop an understanding of orthographic projection.
 * Develop a basic understanding of volume and surface area.

__Isometric drawing__- An isometric drawing is one in which the horizontal lines of an object are represented by lines that form 30° angles with a horizontalline in the picture. [|Orthographic projection]- A view of an object in which points of the object are "projected" onto the picture plane along lines perpendicular to the plane.
 * Vocabulary**:

Left: Orthographic view. Right: Cubes in 3- D.

=Spatial Relationships- Figues, Lines, and Planes in Space; Angles Formed by Planes=


 * Objectives:**
 * Define polhedron.
 * Identify the relationships among point, lines, segments, planes, and angles in three- dimensional space.
 * Define dihedral angle.

__Solid__-Closed spatial figures. //Definition: Polyhedron:// [|Polyhedron]- A polyhedron is a closed spatial figure composed of polygons. __Faces__- The polygons of the plyhedron are called the faces. __Edges__- The intersections of the faces are the edges of the polyhedron. __Vertices__- The vertices of the faces are the vertices of the polyhedron. //Definition: Dihedral Angle:// __Dihedral angle__- A dihedral angle is the figure formed by two half- planes with a common edge. __Face__- Each half- plane is called a face of the angle. __Edge__- The common edge of the half- planes is called the edge of the angle. __Parallel Planes__- Two planes are parallel if and only if they never intersect. __A Line Perpendicular to a Plane__-A line is perpendicular to a plane at a point //P// if and only if it it perpendicular to every line in the plane that goes through //P//. __A Line parallel to a Plane__- A line is not contained in a given plane is parallel to the plane if and only if it is parallel to a line that is in the plane. __Measure of a Dihedral Angle__- The measure of a dihedral angle is the measure of an angle formed by two rays that are on the faces and that are perpendicular to the edge. __Regular Polyhedron__- A regular polyhedron has congruent regular polygons as faces, with the same number of faces meeting at each vertex. Left: Polyhedron. Right: Dihedral angle.
 * Vocabulary**:

=Prisms- Diagonals of Prisms=


 * Objectives**:
 * Define prism, right prism, and oblique prism.
 * Examine the shapes of lateral faces of prisms.
 * Solve problems by using the diagonal measure of a right prism.

[|Prism]- A prism is a polyhedron that consists of a polygonal region and its translated image on a parallel plane, with quadrilateral faces connecting corresponding edges. __Base__- The faces formed by the polygonal region and its image are each called a base of the prism. __Lateral Faces__- The remaining faces, which are quadrilaterals, are called lateral faces of the prism. __Lateral Edges__- The edges of the lateral faces that are not edges of either base are called lateral edges of the prism. __Right Prism__- A right prism is a prism in which all of the lateral faces are rectangles. __Oblique Prism__- An oblique prism has at least on nonrectangular lateral face. __Diagonal of a Right Rectangular Prism__- In a right rectangular prism with dimensions //l// X //w// X //h,// the length of a diagonal is given by //d//= √(//l² + w² +h²//)//.// Left: Oblique building. Right: Hexagonal prism cereal.
 * Vocabulary**:

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= = =Coordinates in Three Dimensions- The Arragement of the Axes; The Octants and the Coordinate Planes; Distance Formula in Three- Dimensions= = =


 * Objectives:**
 * Identify the features of a [|three- dimensional coordinate system,] including the axes, octants, and coordinate planes.
 * Solve problems by using the distance formula in three dimensions.

__Right- Handed System__- A three- dimensional system of coordinates, named from a mnemonic device involving the fingers of the right hand. __Octants-__ One of the eight spaces into which a three- dimensional coordinate system is dividedby the //xy//-, //yz//-, and //xz//-planes. __Coordinate Plane__- A grid formed by two or more coordinatized lines, known as axes, that intersect at right angles at a point known as the origin. __Distance Formula in Three Dimensions__- The distance formula, //d//, between the points (**x**1, **y**1, **z**1) and (**x**2, **y**2, **z**2) is given by //d//=√ (**x**2-**x**1)² + (**y**2-**y**2)² + (**z**2-**z**1)² __Midpoint Formula in Three Dimensions__- The midpoint of a segment with endpoints at (**x**1, **y**1, **z**1) and (**x**2, **y**2, **z**2) is the point __**x**1+**x**2__, __**y**1+**y**2__, __**z**1+**z**2__ ....2.........2.........2
 * Vocabulary**:

Left: Alternative view of a coordinate plane. Right: Example of right handed system.

=Perspective Drawing- Parallel Lines and Vanishing Points; Principles of Perspective Drawing= = =


 * Objectives**:
 * Identify and define the basic concepts of [|perspective drawing.]
 * Apply these basic concepts to create your own perspective drawings.

__Vanishing Point__- The point where parallel lines seem to meet, which is often on the horizon, is known in perspective drawing as the vanishing point. __Theorem: Sets of Parallel Lines__- In a perspective drawing, all lines that are parallel to each other, but not to the picture plane, meet at a single point known as a vanishing point. __Theorem: Lines Parallel to the Ground__- In a perspective drawing, a line that is in the plane of the ground in the drawing and is not parallel to the picture plane will meet the horizon of the drawing. Any line parallel to this line will meet the horizon of the drawing at the same point. Left: A building in two point perspective. Right: A cube in two point perspective.
 * Vocabulary**: