Transcript Discrete Math Point, Line, Plane, Space

Discrete Math
Point, Line, Plane, Space
Desert Drawing to One
Vanishing Point Project
Point
• In Discrete Geometry, a point is a dot.
Point
• The ancient Greeks idealized points as
an exact location, having no size or
shape.
Point
• In Coordinate Geometry: points are
ordered pairs.
Point
• A fourth description of point is of a
node or a vertex in a network.
Point
• Points can make continuous lines.
Point
• Between every point there is always
another point.
Point
• Between every point, there is an infinite
number of points.
Point
• A point cannot be defined or drawn, but
only visualized with a model.
Line
• A line is determined by two points.
Line
• In a plane, a line can: intersect another
line, be parallel to another line, or be
coincident to this line.
Line
• In space, a line can: intersect another
line, be parallel to another line, be
coincident to another line, or be skew to
another line.
Line
• A line cannot be defined or drawn, but
only visualized with a model.
Plane
• A plane is determined by three noncollinear points.
Plane
• When two planes intersect, they form a
line.
Plane
• A plane cannot be defined or drawn, but
only visualized with a model.
Space
• Space is the set of all points.
Space
• When all points in space are collinear,
the geometry is one-dimensional.
Space
• When all points in space are coplanar,
the geometry is two-dimensional (2D)
or plane geometry.
Space
• Other figures, such as spheres, boxes,
cones, and other tangible objects do not
lie in one plane and are threedimensional or 3D. The study of these is
called solid geometry.
Space
• Space cannot be defined or drawn, but
only visualized with a model.
Discrete Geometry
• Models of points:
– Dot matrix printers
– Displays made with LEDs
– Circular metal pipes arranged in hexagonal
prisms
– Some paintings
– Wildflowers in bloom
Models of Points
Models of Points
Models of Points
Models of Points
Models of Points
Euclid’s 5 Postulates
1. To draw a straight line from any point
to any point.
Euclid’s 5 Postulates
2. To produce a finite straight line
continuously in a straight line.
Euclid’s 5 Postulates
3. To describe a circle with any center
and distance.
Euclid’s 5 Postulates
4. That all right angles are equal to one
another.
Euclid’s 5 Postulates
5. That, if a straight line falling on two
straight lines make the interior angles on
the same side less than two right angles,
the two straight lines, if produced
indefinitely, meet on that side on which
are the angles less than the two right
angles.
Euclid to Ptolemy
• When Ptolemy asked if there was an
easier way to learn geometry Euclid
replied: "There is no royal road to
Geometry."
Perspective
• Drawing in Perspective
• Although mathematicians don't often
draw in perspective, the concept and
terminology are important.
Perspective
• A perspective drawing gives a twodimensional object a feeling of depth.
Perspective
• Often one thinks of the artist's or
observer's eye as this vanishing point
and sketches lines of sight to connect
them.
Perspective
• Objects can be drawn in one- two- or
three-point perspective, depending on
how many vanishing points are used.
Perspective
• Parallel horizontal and vertical lines go
to their own vanishing point, depending
on their relationship to each other.
Perspective
• Multiple vanishing points should line up
on the vanishing line which corresponds
with the horizon line at the height of
the observer's eye.
Perspective
• Parallel lines now meet in the distance
at a vanishing point.
Perspective
• Mathematicians typically draw nonperspective drawings, utilizing dashed
or dotted hidden lines to indicate parts
not normally seen.
Perspective
Non - Perspective
Non - Perspective
A Contraction Drawing
Non - Perspective
Anamorphosis
Perspective
Perspective
• Desert Drawing
Perspective
• Desert Drawing
Perspective
• Desert Drawing
Perspective
• Two points vanishing point drawing
Perspective
Perspective
Perspective
Desert Drawing
Desert Drawing
Desert Drawing
Desert Drawing
Desert Drawing
Desert Scene Project
• Using a 11 * 17 inch sheet of white paper
– Lay out a horizon line
– Lay out the road to a left or right vanishing
point
– Use perpendicular lines
– Use parallel lines
– Use points to establish objects
– Use pencil only for a B/W drawing
– Use a ruler at all times for the objects that
need it