Engineering Geometry Engineering geometry is the basic geometric elements and forms used in engineering design. Engineering and technical graphics are concerned with the descriptions of shape, size, and operation of engineered products. The shape description of an object relates to the positions of its component geometric elements in space. To be able to describe the shape of an object, you must understand all of the geometric forms, as well as how they are graphically produced.

Coordinate Space 2-D Space: Cartesian coordinate system: A 2-D coordinate system establishes an origin at the intersection of two mutually perpendicular axes, labeled X (horizontal) and Y (vertical). The origin is assigned the coordinate values of 0,0. X: positive to the right of the origin, and negative to the left. Y: positive above the origin, and negative below Polar coordinate system: Distance from the origin (0,0), and angle measured from the positive X-axis. Distance is always positive. Counterclockwise angle is positive; clockwise is negative.

2-D Cartesian coordinate system

3-D Coordinate Space Cartesian coordinate system Three mutually perpendicular axes (X, Y, and Z) intersect at the origin (0,0,0) The right-hand rule is used to determine the positive direction of the axes. A rectangular prism is created using the 3-D coordinate system by establishing coordinate values for each corner. Cylindrical coordinates locate a point on the surface of a cylinder by specifying a distance and an angle in the X-Y plane, and the distance in the Z direction. Spherical coordinates locate a point on the surface of a sphere by specifying an angle in one plane, an angle in another plane, and one height. Absolute coordinates vs. Relative coordinates World coordinate system vs. Local coordinate system

3-D Cartesian coordinate system

Locating points

Right-hand rule for axes directions

Cylindrical coordinates

Spherical coordinates

Geometric Elements Point, Line, Circle, Arc Parallel lines, perpendicular lines, intersecting lines Tangent line Curved lines: single curved vs. double curved Circle: points equidistant from one point (the center) circumference, radius, chord, diameter, secant, semicircle, arc, sector, quadrant, segment, tangent, concentric circles

Circle definitions

Geometric Elements Conic sections: formed by intersection of a plane with a right circular cone Parabola: set of points equidistant from a fixed point (focus), and a fixed line (directrix) Hyperbola: set of points whose distances from two fixed points (foci) have a common difference Ellipse: set of points whose distances from two fixed points (foci) have a constant sum Polygons and Polyhedrons; prisms and pyramids

Ellipse

HYPERBOLA

PARABOLA

ELLIPSE

CIRCLE

Quadrilaterals

Polygons

Regular polyhedra

Prisms

Pyramids

Design Visualization A dynamic process between the mind, the eyes, and some physical stimulus such as a drawing or an object.

Hand/eye/mind connection

Solid Object Features Edges – lines that represent the boundary between two faces of an object. Faces – areas of uniform or gradually changing lightness and are always bounded by edges. Limiting element – a line that represents the farthest outside feature of a curved surface. Vertex – point where more than two edges meet.

Solid object features

Visualization Techniques Solid Object Combinations and Negative Solids Cutting Planes Normal Rotated about single axis – inclined face Rotated about two axes – oblique face Planes of Symmetry Developments – flattened “skin” of object

Combining solid objects

Removing solid objects

Removing solid objects

Subtracting progressively larger wedges

Subtracting progressively larger pyramids

Additive and subtractive techniques can be used to make a solid geometric form

Normal cutting plane

Cutting plane rotated about single axis

Cutting plane rotated about two axes

Cutting plane rotation

Planes of symmetry

Surface cutting planes

Development

Image plane

Object-image plane orientation

Normal faces

Camera metaphor

Normal face projection

Edge views of normal face

Inclined face projection

Inclined and normal faces

Oblique face projection