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3D Solid Modeling

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Engineering Geometry

Geometry Provides the building blocks of the engineering design process

Coordinate Space

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3D Coordinate Spaces

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Coordinate axes in Multi-View

Only two axes of the three coordinates can be seen in each view

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Right-Hand rule to Determine the Positive Direction of the Axes

Right-hand rule defines the X, Y and Z axes, as well as the positive and negative directions of rotation on each axis

Fist wrapped around the axes, with the thumb pointing to the positive direction of the axis

X: ThumbY: Indexing fingerZ: Middle figure

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Polar Coordinate System

A distance in X-Y plane and an angle from the X axis

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Example I: Polar to Cartesian

D=4.5α=30°

x= D ×

Cos(α)y= D ×

Sin(α)

x=3.897y=2.250

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Example II: Cartesian to Polar

x=4y=6.928

D=8α=60°

4

6.928

⎟⎠⎞

⎜⎝⎛=

+±=

xyatanα

22 yxD

X

Y

D

α

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Cylindrical Coordinate System

A polar system with a Z distance

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Spherical Coordinate System

An angle in one plane, an angle in another plane and one length

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Absolute Coordinates

Points are referenced to the fixed origin

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Relative Coordinates

Starting point, referenced to the origin

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Classification of Geometric Elements

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Representation of Points (examples)

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Representation of Lines (examples)

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Lines Relationships

Parallel Condition Non-parallel Condition

Perpendicular Condition

Intersecting Lines

Tangent Condition

Intersection of Two Planes

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Curved Lines

Path generated by a point moving in a constantly changing direction

Single Curved Line:

All points of the line are in a plane(Circle, Ellipse, Parabola, Hyperbola Spiral, Spline, Involute, Cycloid)

Double Curved Line:

No four consecutive points are in the same plane(Cylindrical helix, 3D Spline)

Regular Curve:

Radius is constant

Irregular Curve:

Radius varies, such as parabola, hyperbola, spline….

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Tangencies

Planar tangents exists when two geometries forms meet at a single point and do not intersect (touch in one and only one point)

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Tangencies in 3D Geometries

Plane touches but not intersect the another surface at one or more consecutive points

Smooth transition between two geometric entitiesNote: Corner between two geometric entities indicates a non-tangent condition

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Tangencies of Line and Surfaces

A line is tangent to a surface, if the line touches the surface only at a single point

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Circles

Diameter is a chord that passes through the center

Circle is a single curved plane with all points at an equal distance from the point called center.

Secant is a line passing through the circle but not the center

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Conics

1.

Ellipse2.

Parabola

3.

Hyperbola

Single curved lines can be described by algebraic equations, loci of points or a section of a cone

Many applications in Engineering and sciences

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ParabolaWhen a plane intersects a right circular cone

parallel to the side (element)

Mathematically, Parabola is defined as a set of points in a plane which are equidistance from a given fixed point, called focus and a fixed line, called

directix.

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Engineering Application of Parabola

Rays coming into the parabola, parallel to the axis are reflected to the focus.

Rays originating at the focus, are reflected out of the parabola parallel to the axis.

Parabola revolved about its axis generates a 3D ruled surface called a paraboloid(Auditorium ceiling reduces reverberation)

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HyperbolaWhen a plane intersects a right circular cone

and makes a smaller angle with the axis than do the elements

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Hyperbola –

Mathematical Definition

Mathematically, Hyperbola is defined as a set of points in a plane whose distances from two fixed points called the foci, in the plane have a constant difference.

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Engineering Application of Hyperbola

Ernest Rutherford discovered that when alpha particles are shot toward the nucleus of an atom, they are repulsed away from the nucleus along hyperbolic paths.

In astronomy, a comet that does not return to the sun follows a hyperbolic path.

Reflecting Telescopes

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When a plane intersects a right circular cone

at an angle to the axis that is greater than the angle between the axis and the sides

Ellipse

Major diameter

is the longest straight line distance between elements and is through the foci.

Minor diameter

is the shortest straight line distance between elements and is through the perpendicular bisector of the major axis.

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Ellipse Templates

A line of sight other than 90 degrees changes the appearance of a circle to an ellipse.

Ellipse templates come in intervals of 5 degrees.The viewing angle relative to the circle, determines the ellipse template to be used.

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Free Form Surfaces

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Free Form Curves

Parametric Equations:1- Splines2-Bezier3- B-Spline4-NURBS

Interpolation vs. Approximation

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Free Form Curves

Spline curve is a smooth free form curve that connects a series of control points

Bezier, B-Spline and NURBS are flexible techniques to create modified smooth curve that passes close to but not through the all of control points.

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AnglesHow to define…

AcuteRightObtuse

Relationships

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Planes

How to define…

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Quadrilaterals

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Regular PolygonsA Polygon with equal sides

12 sides 20 sides10 sides9 sides8 sides

7 sides6 sides5 sides4 sides3 sides

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Triangles

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SurfacesFinite portion of a plane or the outer face of an object bounded by an identifiable perimeter

Ruled Surface is generated by a moving line, called GeneratrixThe path of the moving line is called Directrix

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Classification of SurfacesA) 2D bounded SurfaceB) Straight line generatrix revolved around an axis directrixC) A curved line generatrix revolved around a line, center or a vertexD) A single and double curved transitional surfaceE) Follows no set pattern; more sophisticated math is required

Another ClassificationF) Planar, Single curved or

wrapped surface with a straight line generatrix

G) Can be unfolded or unrolled onto a plane without distortion

H) Can’t be unfolded or unrolled onto a plane without distortion

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Classification of Ruled Surfaces1) Polyhedra2) Single- Curved Surface3) Wrapped Surface

Classification of Single-

Curved Surfaces

1) Polyhedra2) Single- Curved Surface3) Wrapped Surface

An straight line generatrix and path such that any two consecutive position of the generatrix are either parallel, intersecting or tangent to a double curved line (convolute)

Sweeping a curved generatrix along a straight lineOR

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Classification of Cones

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Classification of Cylinders

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Convolutes

A convolute is a single- curved surface generated by a line moving always tangent to a double curved line

Helical convolute is a special case formed by sweep of the line tangent to a helix curve.

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Polyhedra

A symmetrical or asymmetrical geometry with multiple polygonal sides called facet

8 facets

20 facets12 facets

6 facets4 facets

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Polygons Prisms

A polyhedron that has two equal parallel facets

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3-D Modeling

3D computer model of an object

1) Wire frame2) Surface Modeling3) Solid Modeling

B-RepCSG TreeHybrid

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Wire frame

Only vertex and edges Linear and circular edges

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Surface Modeling

Lacking of uniqueness in wire frame

Sweeping generator entities along the director entity

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Surface Modeling

A directrix can be rotated about an axis, between 1 and 360 degrees

Revolved Surface

Complex surfaces by sweeping

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Surface Modeling

Freeform Surface patches using parametric equations

A series of directrix paths and a generatrix curve

Lofting Surface

Generating the surface patches from the boundary curves