Geometric Construction

Geometric Construction

Points and Lines Cartesian Coordinate System Planes Polygons Angles Circles and Ellipse Geometric Constraints Contents

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. Back to Contents

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.

+Y -X -Z +Z +X -Y Cartesian Coordinates 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.

History Fact • 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.

Y 6 5 4 D B 3 2 1 0 A C X 0 1 2 3 4 5 6 7 8 9 Absolute Coordinates A = X3, Y2 B = X4, Y4 C = X7, Y1 D = X8, Y5

Y 6 5 4 D B 3 2 1 0 A C X 0 1 2 3 4 5 6 7 8 9 Relative/Incremental Coordinates Aref zero= X3, Y2 Bref A = X1, Y2 Cref B= X3, Y-3 Dref C= X1, Y4

Y 6 5 4 D B 3 2 1 0 A C X 0 1 2 3 4 5 6 7 8 9 Lines Line AD Line BC

90° 90° 180° 180° Y 6 5 4 0° 0° 270° 270° D A 45° To find this angle we can use an alternative origin. As the 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. 3 2 1 0 To locate point “D”. Use point “A” as the reference point. The “D” Polar Coordinate is 4.25 < 45°. 4.25 is the distance, 45° is the angle the point is located in the coordinate plane. X 0 1 2 3 4 5 6 7 8 9 Polar Coordinates Polar Coordinates work similarly to relative coordinates in that the location of a point is based on the last location point. The difference is that you will locate the next point by distance and the angle the point is located on the coordinate plane.

Right Hand Rule You can use your hand to help orient the coordinate system in a CNC Robotics or CAD application. Make sure you use your right hand! Y Z X Back to Contents

Three points not lying in a straight line • Two parallel lines • Two intersecting lines • A point and a line Geometric Forms • Planes Planes are defined by:

Y Y Y Z Z X X Z X The XY Plane. The XZ Plane The YZ Plane Origin Planes Planes in the origin are identified by the axes that lie on the plane. Back to Contents

Geometric Forms Polygons • 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).

Circle diameter = distance across corners Connect radial lines where the ends intersect the circumference Polygons Inscribed Polygon An inscribed polygon can be constructed by determining the number of sides and the distance across the corners. 8-sided polygon Example: There are 360 in a circle; for an eight-sided polygon divide 360 by 8 (360 8=45 ) to determine the central angle.

Circle diameter = distance across the flats Connect radial lines by drawing line segments tangent to arc segments Polygons Circumscribed Polygon A circumscribed polygon can be constructed by determining the number of sides and the distance across the flats. 8-sided polygon Example: There are 360 in a circle; for an eight-sided polygon divide 360 by 8 (360 8=45 ) to determine the central angle.

Triangle Polygons • Triangle • A triangle is a plane figure bounded by three straight sides. • The sum of the interior angles is always 180°.

Triangle Polygons Equilateral Triangle – All sides equal; all angles equal. Isosceles Triangle – Two sides equal; two angles equal.

Triangle Polygons Right Triangle – Contains one 90 angle. Scalene Triangle – No equal sides or angles.

Quadrilateral Polygons • Quadrilateral • A quadrilateral is a plane figure bounded by four straight sides. • If the opposite sides are parallel, the quadrilateral is also a parallelogram.

Quadrilateral Polygons Parallelograms: Square – All sides equal, four right angles. Rectangle – Opposite sides are equal, four right angles.

Quadrilateral Polygons Parallelograms: Rhombus – All sides equal; Opposite angles are equal. Rhomboid – Opposite sides are equal; Opposite angles are equal.

Quadrilateral Polygons Trapezoid – Two sides parallel. Trapezium – No sides parallel.

Other Polygons Polygons 7 SIDES 5 SIDES 6 SIDES Pentagon Hexagon Heptagon 8 SIDES 9 SIDES 10 SIDES Back to Contents Octagon Nonagon Decagon

Right Angle - Angle equal to 90°. Acute Angle - Angle that is less than 90°. Obtuse Angle - Angle that is greater than 90°. Complementary Angles - Two angles that make up 90°. Supplementary Angles - Two angles that make up 180°. Angle Types

R Strike two arcs, shown here as .625. The arcs can be any size as long as they are equal. Given Angle Strike Arc R any distance. Draw a line from where the .625 arcs intersect to the vertex of the angle. This is the bisector of the angle. The angle is now divided into two equal angles. Bisecting an Angle Back to Contents

Circle Terms Chord is a line that has endpoints at the circumference of a circle The circumference of a circle is the distance around the outside of the circle. To calculate the circumference: Circumference = Dia. X  A circle is a closed curve with all points along the curve an equal distance from a point called the center. Center point The radius of a circle is half the diameter. The diameter is the longest chord in a circle that passes through the center point of a circle. An arc is a portion of the circumference of a circle.

The set of all points in the same plane whose sum of the distances from two fixed points is constant. Ellipse The sum of the distances of 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.

These two lines are parallel. The lines are also representing the symbol for parallel. These two lines are perpendicular and represent the symbol for perpendicular. Geometric Constraints Parallel - Lines that are equal distance from each other at each point along their length. Perpendicular - Lines that are 90° from one another.

Geometric Constraints Horizontal - 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.

Figure B Figure A Geometric Constraints Tangent - A line or arc that has one point in common with 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.

These circles and the arcs share the same center point. Geometric Constraints Concentric - Circles or arcs that share the same center point. Coincident - Points that share the same location on the coordinate plane. Points may also be parts of arcs or curves.

Collinear - Lines that if projected at each other will become the same line. Collinear lines Geometric Constraints 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 given the same magnitude. Back to Contents

Geometric Construction