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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.
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 three-dimensional or 3D. The study of these is called solid geometry.
Discrete Geometry
• Models of points:– Dot matrix printers– Displays made with LEDs– Circular metal pipes arranged in
hexagonal prisms– Some paintings–Wildflowers in bloom
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
• 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
• Mathematicians typically draw non-perspective drawings, utilizing dashed or dotted hidden lines to indicate parts not normally seen.
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