The shortest distance between two points is a curve

Contrary to what you learned in 7th grade geometry, a straight line is not the shortest distance between two points. A flowing curve pulls your eye along sweetly, whereas a straight line often halts the eye.

I just finished proofing an article on curvature for the February issue of Popular Woodworking Magazine, and another article for the annual SAPFM journal on how to lay out a Swan Neck Pediment. Both articles share some simple and quick layout methods that will help you visualize curves. This weekend, stop by the SAPFM booth at the Midwest WIA where I’ll be demonstrating how to lay out a Swan’s neck or Scrolled pediment. Even if you have absolutely no interest in this traditional form, it’s a marvelous exercise to help improve your vision and boost your confidence working with curves. Prepare to get dazzled!  I’ll be about all day Saturday, so if you miss the demo or have a design question, don’t hesitate to pull me aside. I’d love to see what you are up to.

George R. Walker

About walkerg

Woodworker and writer
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9 Responses to The shortest distance between two points is a curve

  1. mike siemsen says:

    It is funny you mention what we learned in geometry class. My High School geometry instructer told us that the shortest distance between to points is a curve. He was a naval navigator in a former life. Something about the great circle.

  2. chuck nickerson says:

    Inquiring minds would like to know. In the drawing, I see that the point of inflection is suggested by the relative size of the two circles. Is there a rule that suggests the height of each equilateral triangle?

    • William Duffield says:

      Here is an explanation of George’s construction, as I see it. It starts with two arbitrary circles. The first artistic decision is to have the curve meet the circles tangentially, 180° apart, in this case at the top of the smaller circle and at the bottom of the larger circle. The first line connects these intersection points. The second and third lines (diameters) pass through those points and the centers of the respective circles. The second artistic decision is the proportions of the lengths of the two arcs of the connecting curve. In this case it is 2:1, which happens to be different than the proportions of the sizes of the circles. Hopefully George will explain why he made the artistic decisions. While we are waiting, I’ll continue with the construction: Trisect the first line. Construct a perpendicular through the trisection point nearer the large circle. It is not shown in the drawing, but the center of the major arc is the intersection of this perpendicular with the diameter extended through the larger circle. Draw a line from this center through the trisection point nearer the small circle. Where this line intersects the diameter drawn through the smaller circle is the center of the minor arc.

      I think when George says “shortest” he really means “quickest”, but a highway engineer might even disagree with the efficiency of this construction.

  3. chuck nickerson says:

    Just saw it. Putting dividers to a work monitor produced all sorts of amused looks.

  4. luke b says:

    i dont get it, how can this possible? is this a metaphor or something?

  5. Mark says:

    I really enjoyed reading this although im not fully sure I understand *confused face – Im quite intrigued to find out more but i was also wondering about the height of the triangles and whether there is a rules behind that?

    • walkerg says:

      Concerning the heights of the triangles, there are no rules. They are simply the outcome of dividing the straight line or chord into equal parts and creating a major and minor curve. In this case the chord is divided into three parts with one part given to the minor and two to the major. The only rule or logic is that by using simple ratios to govern the curves, you have a large number of possibilities to play with.


      • William Duffield says:

        And the total length of the curve ends up the same, no matter how you proportion the chord into major and minor curves. You could even use the golden ratio.

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