Transcript Geometric Construction - Lancaster High School

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.
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.
Cartesian Coordinates
+Y
-X
The origin is the place where X,
The
X, ZY,are
andallZ equal
axes are
number lines
Y,
and
to zero.
that are oriented as you see here.
Placing the axis in this manner the
user can locate points in space
-Z
+Z
+X
-Y
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
Y
6
D
5
A = X3, Y2
B
4
3
B = X4, Y4
C = X7, Y1
A
2
D = X8, Y5
C
1
0
0 1
2
3
4
5
6
7
8
9
X
Relative/Incremental Coordinates
Y
6
A ref zero= X3, Y2
D
5
B ref A = X1, Y2
B
4
C ref B= X3, Y-3
3
A
2
D ref C= X1, Y4
C
1
0
0 1
2
3
4
5
6
7
8
9
X
Lines
Y
6
D
5
Line AD
B
4
3
A
2
C
1
Line BC
0
0 1
2
3
4
5
6
7
8
9
X
90°
Polar Coordinates
Y
180°
D
0°
6
5
90°
4
3
270°
A
45°
similarly to
To find thisPolar
angleCoordinates
we can usework
an alternative
origin. As
0°
relative
coordinates
in
that
the
location
the reference point changes, this new origin will be placed
of a point
is based
on
lastbe
location
onlocate
the point
and
the
angle
canthe
then
measured.
To
point
“D”.
Use
point
“A”
as
the
reference
point.
The
difference
is
that
you
will
locate
Notice
that
the
angle
is measured
in a< counter
point.
The
“D”
Polar
Coordinate
is
4.25
45°.
4.25
the next point
by
distance
and
the
angle
clockwise
direction.
is thethe
distance,
is theon
angle
the point isplane.
located
point is45°
located
the coordinate
in the coordinate plane.
180°
2
1
270°
0
0 1
2
3
4
5
6
7
8
9
X
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!
Z
Y
X
Back to Contents
Geometric Forms

Planes
Planes are defined by:
Three points not lying in a straight line
Two parallel lines
Two intersecting lines
A point and a line
Origin Planes
YY Y
Planes in the origin are
identified by the axes that
lie on the plane.
ZZZ
XX
X
TheThe
XY Plane
Plane.
The
YZ
XZ
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).
Polygons
Inscribed Polygon
An inscribed polygon can be
constructed by determining the
number of sides and the distance
across the corners.
Circle diameter = distance across corners
8-sided polygon
Example:
Connect radial lines where the ends
intersect the circumference
There are 360 in a circle; for an eight-sided polygon divide
360 by 8 (360 8=45 ) to determine the central angle.
Polygons
Circumscribed Polygon
A circumscribed polygon can be
constructed by determining the
number of sides and the distance
across the flats.
Circle diameter = distance across the
flats
8-sided polygon
Example:
Connect radial lines by drawing line
segments tangent to arc segments
There are 360 in a circle; for an eight-sided polygon divide
360 by 8 (360 8=45 ) to determine the central angle.
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.
Polygons
Quadrilateral
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
Quadrilateral
Parallelograms:
Square – All sides equal, four right angles.
Rectangle – Opposite sides are equal,
four right angles.
Polygons
Quadrilateral
Parallelograms:
Rhombus – All sides equal; Opposite
angles are equal.
Rhomboid – Opposite sides are equal;
Opposite angles are equal.
Polygons
Quadrilateral
Trapezoid – Two sides parallel.
Trapezium – No sides parallel.
Other
Polygons
Polygons
5 SIDES
6 SIDES
7 SIDES
Pentagon
Hexagon
Heptagon
8 SIDES
9 SIDES
10 SIDES
Back to Contents
Octagon
Nonagon
Decagon
Angle Types
Acute Angle - Angle that is
less than 90°.
Complementary Angles Two angles that make up
90°.
Obtuse Angle - Angle
that is greater than 90°.
Right Angle - Angle
equal to 90°.
Supplementary Angles Two angles that make up
180°.
Bisecting an Angle
R
Given Angle
Strike Arc R any
distance.
Strike two arcs, shown here as .625.
The arcs can be any size as long as
they are equal.
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.
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
Adistance
circle is around
a closedthe
curve
withofallthe
outside
points
the curve
an equal
Center
point
circle.
Toalong
calculate
the circumference:
distance
from a point
Circumference
= called
Dia. Xthe

center.
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.
Ellipse
The set of all points in the same
plane whose sum of the distances
from two fixed points is constant.
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.
Geometric Constraints
Parallel - Lines that are equal distance from
each other at each point along their length.
These two lines are parallel. The lines are
also 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 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.
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.
Figure A
Figure B
Geometric Constraints
Concentric - Circles or arcs that share the same center
point.
These circles and the arcs 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.
Geometric Constraints
Collinear - 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 given
the same magnitude.
Back to Contents