Geometric Construction

Contents Points and Lines Cartesian

Coordinate System

Planes Polygons

Angles Circles and Ellipse Geometric

Constraints

Geometric Forms Points

Points are used to indicate locations in space. Points are considered to have no height, width or

depth. A point can be defined as a set of coordinates

(x,y) on the Cartesian plane. Lines

A straight line is the shortest distance between two points.

Lines are considered to have length, but no other dimension such as width or thickness.

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Cartesian Coordinate System In order to locate objects in space,

the Cartesian Coordinate System is used. This system not only helps to locate objects, but allows for sizes to be made from those locations.

Cartesian Coordinates

+X

+Y

+Z

-X

-Y

-Z

The X, Y, and Z axes are number lines that are oriented as you see here. Placing the axis in this manner the

user can locate points in space

The origin is the place where X,Y, and Z are all equal to zero.

Cartesian Points Absolute Coordinates

Points that are defined by absolute coordinates refer to the origin for their numeric value. The point is identified by the absolute X and Y distance from zero.

Relative or Incremental Coordinates Points that are defined by relative coordinates

reference the previous point on the Cartesian plane. The final point is identified by the distance from the last point referenced.

Renee Descartes (1596 – 1650) was the French philosopher and mathematician for whom the Cartesian Coordinate System is named.

The Cartesian Coordinate System made it possible to represent geometric entities by numerical and algebraic expressions

Descartes coordinate system remains the most commonly used coordinate system today for identifying points.

Absolute Coordinates

X

Y

0 1 2 3 4 5 6 7 8 9

3

2

1

0

6

5

4

A = X3, Y2

B = X4, Y4

C = X7, Y1

D = X8, Y5

A

B

D

C

Relative/Incremental Coordinates

X

Y

0 1 2 3 4 5 6 7 8 9

3

2

1

0

6

5

4

A

B

D

C

A ref zero= X3, Y2

B ref A = X1, Y2

C ref B= X3, Y-3

D ref C= X1, Y4

X

Y

0 1 2 3 4 5 6 7 8 9

3

2

1

0

6

5

4

A

D

B

C

Lines

Line AD

Line BC

Polar Coordinates

X

Y

0 1 2 3 4 5 6 7 8 9

3

2

1

0

6

5

4Polar Coordinates work similarly to

relative coordinates in that the locationof a point is based on the last location

point. The difference is that you will locatethe next point by distance and the angle

the point is located on the coordinate plane.

0°

90°

180°

270°

To find this angle we can use an alternative origin. Asthe reference point changes, this new origin will be placed

on the point and the angle can then be measured.Notice that the angle is measured in a counter

clockwise direction.

A

D

0°

90°

180°

270°

To locate point “D”. Use point “A” as the reference point. The “D” Polar Coordinate is 4.25 < 45°. 4.25is the distance, 45° is the angle the point is located

in the coordinate plane.

45°

Right Hand Rule

X

YZ

You can use yourhand to help orientthe coordinate system in a CNCRobotics or CADapplication. Make sure you use your right hand!

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PlanesPlanes are defined by:

Geometric Forms

Three points not lying in a straight lineTwo parallel lines

Two intersecting lines

A point and a line

Origin Planes

Planes in the origin areidentified by the axes that lie on the plane.

X

Y

Z

The XY Plane.

Y

Z X

The XZ Plane

Y

XZ

The YZ PlaneBack to Contents

A polygon is any closed plane, geometric figure with three or more sides or angles.

Polygons can be inscribed (drawn within a circumference) or circumscribed (drawn around a circumference).

Geometric Forms

Polygons

PolygonsAn inscribed polygon can be constructed by determining the number of sides and the distance across the corners.

8-sided polygon

Circle diameter = distance across corners

Example:There are 360 in a circle; for an eight-sided polygon divide 360 by 8 (360 8=45 ) to determine the central angle.

Inscribed Polygon

Connect radial lines where the ends intersect the circumference

PolygonsA circumscribed polygon can be constructed by determining the number of sides and the distance across the flats.

8-sided polygon

Circle diameter = distance across the flats

Example:There are 360 in a circle; for an eight-sided polygon divide 360 by 8 (360 8=45 ) to determine the central angle.

Connect radial lines by drawing line segments tangent to arc segments

Circumscribed Polygon

Polygons Triangle

Triangle

•A triangle is a plane figure bounded by three straight sides.

•The sum of the interior angles is always 180°.

Polygons Triangle

Equilateral Triangle – All sides equal; all angles equal.

Isosceles Triangle – Two sides equal; two angles equal.

Polygons Triangle

Right Triangle – Contains one 90 angle.

Scalene Triangle – No equal sides or angles.

PolygonsQuadrilateral

Quadrilateral

•A quadrilateral is a plane figure bounded by four straight sides.

•If the opposite sides are parallel, the quadrilateral is also a parallelogram.

Polygons

Square – All sides equal, four right angles.

Rectangle – Opposite sides are equal,four right angles.

Quadrilateral

Parallelograms:

Polygons

Rhombus – All sides equal; Opposite angles are equal.

Rhomboid – Opposite sides are equal; Opposite angles are equal.

Quadrilateral

Parallelograms:

Polygons

Trapezoid – Two sides parallel.

Trapezium – No sides parallel.

Quadrilateral

PolygonsOther

Polygons

Pentagon Hexagon Heptagon

Octagon Nonagon Decagon

5 SIDES 6 SIDES 7 SIDES

9 SIDES8 SIDES 10 SIDES

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Angle Types

Acute Angle - Angle that isless than 90°.

Obtuse Angle - Anglethat is greater than 90°.

Right Angle - Angle equal to 90°.

Complementary Angles - Two angles that make up 90°.

Supplementary Angles - Two angles that make up180°.

Bisecting an Angle

Given Angle

R

Strike Arc R any distance.

Strike two arcs, shown here as .625.The arcs can be any size as long asthey are equal.

Draw a line from where the .625 arcs intersectto the vertex of the angle. This is the bisectorof the angle. The angle is now divided into two equal angles.

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Circle Terms

A circle is a closed curve with allpoints along the curve an equaldistance from a point called the

center.

Center point

Chord is a line that has endpointsat the circumference of a circle

The diameter is the longest chord in a circle that passes through the

center point of a circle.

The radius of a circle is half the diameter.

An arc is a portion of the circumference of a circle.

The circumference of a circle is thedistance around the outside of the

circle. To calculate the circumference: Circumference = Dia. X

EllipseThe set of all points in the same

plane whose sum of the distances from two fixed points is constant.

The sum of the distancesof the black lines equals the sum of the distances

of the red lines. Back to Contents

Geometric Constraints When making solid models,

constraints are necessary to produce parts of exact shapes and sizes. To make a part parametric it is necessary to use as many geometric constraints as possible. The next set of slides will show what that geometry is.

Geometric Constraints

Parallel - Lines that are equal distance fromeach other at each point along their length.

These two lines are parallel. The lines arealso representing the symbol for parallel.

Perpendicular - Lines that are 90° from one another.

These two lines are perpendicular and represent the symbol for perpendicular.

Geometric ConstraintsHorizontal - A line is horizontal when it is parallel to the horizon. In solid modeling, the line is also parallel in the horizontal projection plane and will appear true length.

Vertical - A line is vertical when it is perpendicular to the horizon. This line will be parallel to the front and profile projection planes.

Geometric ConstraintsTangent - A line or arc that has one point in commonwith an arc. If a line is tangent with a circle(Figure A), the line will be perpendicular with a line drawn from the point of tangency through the center point of the arc. If two arcs are tangent (Figure B), a line drawn between the centers will intersect at the point of tangency.

Figure A Figure B

Geometric ConstraintsConcentric - Circles or arcs that share the same centerpoint.

These circles and the arcs share the samecenter point.

Coincident - Points that share the same locationon the coordinate plane. Points may also be parts of arcs or curves.

Geometric ConstraintsCollinear - Lines that if projected at each other will become the same line.

Collinear lines

Coplanar - Two or more objects that sit in the same plane.

Fixed Point - A point that has been forced to stay in one location in space.

Equal - Two or more lines, arcs, or circles that are giventhe same magnitude. Back to Contents