loco loci - bible studies by steve loci4.pdf · loco loci loco loci demonstrating with geogebra dr....
TRANSCRIPT
6/11/2015
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Loco Loci
Loco Loci
Demonstrating
with Geogebra
Dr. Steve Armstrong
Dept. of Mathematics (Retired)
LeTourneau University
This is not the
Locus you are
looking for
Definition
• Locus of points ⇔ a curve
– Formed by all the points
– Satisfying a particular equation of the
relation between coordinates.
• The plural of locus is loci.
Conic Sections
• Each can be described as a locus of points
Circle
• Simplest locus
• A circle is a set of points,
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Parabola
• Quadratic equations
• Parabola
– All points equidistant from a fixed point and a
fixed line
– Fixed line, k,
called
directrix
Parabola
Parabola Reflective Property Parabola Reflective Property
Parabola Reflective Property Parabola Reflective Property
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Reflection in Circular Arc
• Move focus to get different parallel signals.
Ellipse
• Defined as set off all points, P, sum of whose
distances from F1 and F2 is a constant
Ellipse Ellipse Reflective Property
Ellipse Reflective Property
John Quincy Adams eavesdropped on senate members
Hyperbola
• Defined as set off all points, P, difference of
whose distances from F1 and F2 is a constant
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Hyperbola Hyperbola Reflective Property
Cycloid
• Definition: the curve traced by a point on the
rim of a circular wheel as the wheel rolls along
a straight line
Animated Cycloid
• Note
– Circle not actually “rolling”
– Distance of center from 0
– Arc calculated to place point
Animated Cycloid Animated Cycloid
• Use formula
• Angle θ places point
• Use slider to move center
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Cycloid on Train Wheel
• Consider the construction of a train wheel
– Outer circumference is below the rail
Cycloid on Train Wheel
• Agree or disagree?
– As the train moves forward …
– A percentage of the train is moving backwards
Cycloid on Train Wheel
• Watch the point on the rim in relation to the
center of the wheel.
Cycloid on Train Wheel
• Watch the point on the rim in relation to the
center of the wheel.
• This might be easier to see.
Cycloid on Train Wheel
• Let horizontal axis be time
• Let of point on rim show velocity
Cycloid on Train Wheel
• Horizontal rate of change on rim
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Spirograph Loci
• Circle rolling on the outside of a circle
Spirograph Loci
• Pattern depends on ratio of circle radii
Spirograph Loci
• Pattern depends on ratio of circle radii
Spirograph Loci
• Pattern depends on ratio of circle radii
Spirograph Loci
• Circle rolling inside the first circle.
Spirograph Loci
• Again, pattern depends on ratio of radii
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Spirograph Loci
• Again, pattern depends on ratio of radii
Spirograph Loci
• Again, pattern depends on ratio of radii
Not Your Normal Spirograph
• Consider circle rolling on an ellipse …
Animating the Point
• Use formula
• Angle θ places point
• Use slider to move center
Animating the Point Animating the Point
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Back to the Ellipse
• Calculating for rotation
Not Your Normal Spirograph
• Calculating for rotation
Not Your Normal Spirograph
• Avoid complicated calculations
• Create lookup table
• Use distance formula
Cumulative Distances
Not Your Normal Spirograph
Not Your Normal Spirograph
• Circle inside the ellipse
Harmonograph
• Two pendulums
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Harmonograph
• Sample drawings
Harmonograph
• Decaying sine and cosine functions
• Graphed as parametrics
Harmonograph
• Geogebra result
Pintograph
Pintograph Pintograph
• Cardiod
and triangle
• Points travel
on shape
to drive
linkage
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Adjustable Four Bar Linkage Geogebra for You
• Free at http://www.geogebra.org
– Available for iPad,
Kindle, Android
• Many free and
interactive materials
at http://geogebratube.org/
Bug on Pendulum Circle on Lissajous Curve
Questions
Loco Loci
Demonstrating
with Geogebra
Dr. Steve Armstrong
Dept. of Mathematics (Retired)
LeTourneau University