2brosig

=3D Shapes=


 * Objectives:**
 * 1) Draw cubes in three dimensions on isometric dot paper.
 * 2) Understand orthographic projection.
 * 3) Understand volume and surface area & apply these concepts to three-dimensional shapes.

[|isometric drawing] - lines that form 30° angles represent the horizontal lines of an object (drawing in three dimensions). [|*Link to isometric dot paper*] [|orthographic projection] - a two-dimensional graphic representation of an object from six different views: front, back, top, bottom, left and right.
 * Definitions:**


 * Key Concepts:**
 * Find surface area** of three-dimensional shapes by counting the total number of visible (sometimes hidden) faces of a cube in an isometric drawing. Measured in units³.
 * Find volume** of a three-dimensional shape by counting the total number of cubes in an isometric drawing. Measured in units².


 * Sample Problems:**



__Isometric cube__ a. find the number of faces in the figure above.

__Three-dimensional cubes on isometric dot paper.__ a. find volume of figure above. b. find surface area of figure above.

=**Spatial Relationships**=


 * Objectives:**
 * 1) Find definition of //polyhedron.//
 * 2) Find the relationships between points, lines, segments, planes, and angles in a three dimensional shape.
 * 3) Find the definition of //dihedral angle.//

[|polyhedron-]** a 3D figure composed of polygons. (see above.)
 * Definitions:
 * **faces of a polyhedron-** the polygons that make up a polyhedron
 * **edges of a polyhedron-** intersections of the faces
 * **vertices of a polyhedron-** vertices of the faces


 * parallel planes-** two planes are parallel if and only if they never intersect.
 * a line perpendicular to a planes-** at point P, a line is perpendicular to a plane, if and only if it is perpendicular to every line in the plane that passes through the point.
 * a line parallel to a plane-** a line that is not contained in given plane is parallel to the plane if and only if it is parallel to a line contained in the plane.
 * [|dihedral angle-]** formed by two half-planes with a common edge.
 * **face of the angle-** each half plane
 * **edge of the angle**- common edge of the half-planes.


 * measure of a dihedral angle-** the measure of a dihedral angle is the measure of the angle formed by two rays that are on the faces and are perpendicular to the edge.


 * Sample Problems:**

=Prisms=


 * Objectives:**
 * 1) Find the definitions of prism, right prism, and oblique prism.
 * 2) Shapes of lateral faces of prisms.
 * 3) Use diagonal measure of a right prism to solve problems.

[|prism] -** a 3D polyhedron that contains 1 polygonal region, which is translated on a parallel plane, with faces connecting edges of polygonal region and the translated image.
 * Definitions:
 * **bases of a prism-** polygonal region and it's translated image.
 * **lateral faces of a prism-** remaining quadrilateral faces which connect polygonal region and it's translated image
 * **lateral edges of a prism-** edges of lateral faces that are not edges of bases.
 * right prism-** all the lateral faces are prisms.
 * [|oblique prism-]** non rectangular lateral faces (at least one).
 * diagonal of a polyhedron-** segment whose endpoints are vertices of two different faces of the polyhedron.

In a prism (right rectangular) with dimensions //l// × //w// × //h//, length of diagonal is given by **d= //l//² + //w//² + //h//²**
 * Key Concepts:**
 * Finding the diagonal of a right rectangular prism-**

=Coordinates in three dimensions=


 * Objectives:**
 * 1) Identify characteristics of a 3D cooxrdinate system- axes, octants, and coordinate planes.
 * 2) Use distance formula to solve problems in three dimensions.


 * Definitions:**
 * right-handed system-** arrangement of axes in a three-dimensional **x,y,z** format (in contrast to the traditional **x,y** format.).
 * octants-** the eight spaces the **x,y,z** plane is divided into. (Front, back, top, bottom, left, right)
 * coordinate plane-** determined by each pair of axes. Includes the xy-plane, xz-plane, and yz-plane

negative number (-5, -3, etc.): back negative number: left negative number: bottom
 * Key Concepts:**
 * To Find Octants-**
 * x**- positive number (3, 4, etc.): front
 * y**- positive number: right
 * z**- positive number: top

In the **xy-plane**, the z-coordinate of every point is 0. In the **xz-plane**, the y-coordinate of every point is 0. In the **yz-plane**, the x-coordinate of every point is 0.

The Distance between points (//x¹, y¹, z¹//) and (//x², y², z²//) is found by **d= √((x²- x¹)² + (y² - y¹)² + (z² - z¹)²) .**
 * Distance Formula in 3D-**

The midpoint of a segment is found by **(x¹ + x² ÷ 2, y¹ + y² ÷ 2, z¹ + z² ÷ 2)**
 * Midpoint Formula in 3D**

Name the octant, axis, or coordinate plane for each.
 * Sample Problems:**


 * 1. (9, -3, 6)**

__Solution:__ 9: positive x = front -3: negative y = left 6: positive z= top //Front, Left, Top//


 * 2. (8, 5, 0)**

__Solution:__ No z-axis is present. //xy-plane//


 * 3. (0, 0, -4)**

__Solution:__ No x- or y- axes present. //z- axis//


 * Find the Distance Between Points**


 * 1. R (8, 5, 7) and S (4, 3, 5)**

__Solution:__ RS= √((8-4)² + (5-3)² + (7-5)²) √((4)² + (2)² + (2)²) √(16 + 4 + 4) //√(24) or about 4.9//


 * 2. X (2, 3, 5) and Y(12, 7, 10)**

__Solution__ XY= √((12-2)² + (7-3)² + (10-5)²) √((6)² + (4)² + (5)²) √(36 + 16 + 25) //√(77) or about 8.8//


 * Find the Midpoint**


 * 1. (10, 3, 8) and (1, 2, 8)**

__Solution:__ (10 + 1 ÷ 2, 3 + 2 ÷2, 8 + 8 ÷ 2) (11 ÷ 2, 5 ÷2, 16÷2) //(5.5, 2.5, 8)//

=Perspective Drawings=


 * Objectives:**
 * 1) Identify basic concepts of perspective drawing.
 * 2) Apply concepts of perspective drawing to create own perspective drawings.


 * Definitions:**
 * perspective drawing-** a drawing in which parallel lines, in order to show perspective, meet at a //vanishing point.// (see above)
 * vanishing point-** the point where parallel lines seem to met, often called the horizon.
 * [|single point perspective drawings-]** a drawing that has one vanishing point.
 * two-point perspective drawings-** a drawing with two vanishing points.

If lines meet at a single point in a perspective drawing, then the lines are parallel to each other, but not the picture plane. In a perspective drawing, any line that is parallel to a line parallel to the ground will meet the horizon of the drawing. A line that is in the plane of the ground and not parallel to the picture plane will also meet the horizon.
 * Key Concepts:**
 * Theorem 1.1:**
 * Theorem 1.2:**


 * [|Drawing a one-point perspective drawing]**
 * 1) Draw a square and horizontal line above, below, or in the middle of the square (this is the horizon). Mark a point on the horizon (vanishing point).
 * 2) From the each corner of the square draw light lines connecting to the vanishing point on your horizon.
 * 3) Draw smaller square with lines touching the lines you drew in the last step.
 * 4) Erase lines that extend past the smaller cube, including the horizon and vanishing point.

//For a two-point perspective drawing, you should end up with something that looks a little like this before you erase pencil lines. . .//
 * [|Drawing a two-point perspective drawing]**
 * 1) Draw a vertical segment. This will be the front edge of your cube. Draw a horizontal line (horizon) above this segment and place two points on this horizon line (vanishing points), one on each side of the vertical line.
 * 2) Draw light lines from endpoints of vertical segment to the vanishing point on both sides of vertical segment.
 * 3) Draw two vertical segments on both sides of the present vertical segment.
 * 4) Connect the ends of new vertical segments to the vanishing point diagonal from it.
 * 5) Erase lines that extend beyond edges of the cube.