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 non- collinear 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 three- dimensional 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

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 two- dimensional 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 non- perspective drawings, utilizing dashed or dotted hidden lines to indicate parts not normally seen.

Perspective

Non - Perspective

A Contraction Drawing

Non - Perspective

Anamorphosis

Perspective

Desert Drawing

Perspective Desert Drawing

Perspective Desert Drawing

Perspective Two points vanishing point drawing

Perspective

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