One dimension

1
One dimension

• Lines, circles, and curves
• According to Euclid, a line is breadthless
  length, that is, it possesses one-dimensionality,
  but no other dimension.
• Actual lines
• may end (line segments)
• have some form of width (especially in Paint and
  Adobe Illustrator)

2
Slope

• Given a line, its slope is defined as rise over
  run, or

3
Examples

• Find the slope of the following line segments
• the line segment from (1,4) to (3,10)
• the diagonals of a golden rectangle (long side
  horizontal)
• a horizontal line
• a vertical line

4
Lines and angles

• Reminder a circle is broken up into 360 degrees.
• Given any line, we can find the angle it makes
  with the horizontal.
• Convention we use negative angles for downward
  sloping lines
• Example diagonals of a square.

5
The tangent function

• Given the angle ? a line makes with the
  horizontal, tan ? gives the slope of the line.
• Given a slope m, tan-1 m gives the angle the line
  makes with the horizontal.
• Examples discuss tan 45, tan 90,
  tan-1(1), and tan-1(-?).

6
Aesthetic lines

• De Stijl
• Theo van Doesburg
• Piet Mondrian
• Henrys aesthetic protractor
• the harmonious angles were those that evenly
  divided 360 (or were multiples of those angles)

7

• Seurats La Chabut as diagrammed in Homers
  Seurat and the Science of Painting
• Supposedly constructed using Henrys aesthetic
  protractor

8
Kazimir Malevich

• John Milnors Kazimir Malevich and the Art of
  Geometry argues that Malevich intentionally used
  angles that were multiples of 9.
• Peasant Woman with Buckets (1912)
• Find important lines on the figure, find their
  slope, and use the tan-1 function to determine
  the angles.

9
Circles

• Distance definition a circle is the set of
  points a fixed distance from a fixed points.
• We say the circle is the locus of these points.
• Function definition the circle is the shape
  traced out by the parametric equation (cos ?, sin
  ?) as ? ranges from 0 to 360 degrees.
• Example show this with a calculator.

10
Example

• On graph paper, determine what shapes you create
  when you trace out the following parametric
  equations
• (cos ?, 1 sin ?)
• (2cos ?, sin ?)
• Note that you can check your answer by using this
  online parametric grapher.