History of Maths

Simona Riva, 08/giu/2024

Collection of worksheets about History of Mathematics: classical constructions, curves as loci of points, envelopes...

Sommario

  1. Classical Curves - Loci
    1. Astroid - A Classical Construction
    2. Limaçon / Cardioid as a Locus - A Classical Construction
    3. Rose Curve - A Classical Construction
    4. Witch of Agnesi - A Classical Construction
    5. Locus Related to Descartes' Folium
  2. Envelopes
    1. Ellipse as Envelope - Correspondence Method
    2. Ellipse, Hyperbola and Circle as Envelopes
    3. "Compressed" Astroid as Envelope
  3. Other Constructions
    1. Ogee Arch - Construction
  4. Plane Surfaces
    1. Pelecoid
  5. 3D Curves and Surfaces
    1. Viviani's Window
  6. ... and more
    1. Parabolas, Lines and Nomograms

1.

Classical Curves - Loci

  • 1. Astroid - A Classical Construction
  • 2. Limaçon / Cardioid as a Locus - A Classical Construction
  • 3. Rose Curve - A Classical Construction
  • 4. Witch of Agnesi - A Classical Construction
  • 5. Locus Related to Descartes' Folium

1.1.

Astroid - A Classical Construction

Classical construction of the astroid as a locus.[br][br]To generate the locus, move point [math]A[/math] along the circle, or use the animation button on top right of the app.
– Simona Riva

1.2.

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1.3.

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1.4.

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1.5.

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2.

Envelopes

  • 1. Ellipse as Envelope - Correspondence Method
  • 2. Ellipse, Hyperbola and Circle as Envelopes
  • 3. "Compressed" Astroid as Envelope

2.1.

Ellipse as Envelope - Correspondence Method

Consider the function [math]f\left(x\right)=\frac{1}{x}[/math] . [br]Represent domain and range of this function respectively over two parallel reference axes.[br]The segments that join each point of the domain with the corresponding point of the range are the envelope of an ellipse.[br][br]Use the sliders [i][color=#a64d79]distance[/color][/i] and [math]O'[/math] to change the distance between axes and the relative position of the Origin points on the two axes.
– Simona Riva

2.2.

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2.3.

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3.

Other Constructions

  • 1. Ogee Arch - Construction

3.1.

Ogee Arch - Construction

Step-by-step construction of an [url=https://en.wikipedia.org/wiki/Ogee]ogee arch[/url], architectural structure of Indian origin - third century B.C.
– Simona Riva

4.

Plane Surfaces

  • 1. Pelecoid

4.1.

Pelecoid

– Simona Riva

5.

3D Curves and Surfaces

  • 1. Viviani's Window

5.1.

Viviani's Window

– Simona Riva

6.

... and more

  • 1. Parabolas, Lines and Nomograms

6.1.

Parabolas, Lines and Nomograms

Before the invention of the calculator, people always sought methods or tools to perform calculations easily, such as the abacus, which dates back to ancient times (21st century BC), or the slide rule in more recent times (17th century AD).[br][br]The nomogram (1884 - Philbert Maurice d'Ocagne), a simple example of which is shown below, is a graphical calculation tool instead. [br][br]A nomogram generally consists of three scales (in our case, the two branches of the parabola and the y-axis): the line segment connecting two points on the outer scales intercepts on the third scale the result of the operation the tool was designed for.[br][br]Can you figure out what the nomogram in the app is used for?
A Question...
Do you think that if we used points on the parabola with rational [i]x[/i]-coordinates instead of integer ones, we would still get the expected result of the operation?[br]Explain the reason for your answer.
– Simona Riva

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