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A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan [email protected] IREM of Toulouse 14 de T 3 Europe Symposium Oostende 22-23/08/201
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A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan [email protected] IREM.

Dec 28, 2015

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Page 1: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

A Dynamic Approach of Analytic Geometry in 3D

with TI N’Spire Enhancing an Experimental Process of Discovery

Jean-Jacques [email protected]

IREM of Toulouse

14de T3 Europe Symposium Oostende 22-23/08/2011

Page 2: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

INTRODUCTION

Representing 3D objects in 2D with two parallel perspectives

Page 3: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

The « cavaliere » and the « military » perspectives

« Cavaliere » perspective « Military » perspective

PC.cg3 PM.cg3

Page 4: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Theses perspectives with dynamic numbers in the « Geometry » application of TI

N’Spire

Paper1 problem 1

Page 5: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

An example of representation Circles in base planes

Paper1 problem 1

Page 6: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Another example using dynamic numbers: Dynamic coordinates for movable points

Paper 1 problem 2

Page 7: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

PART 1 CYLINDERS and CONES

Their representations in« cavaliere » and « military »

perspectives

Page 8: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

With traces and loci

Paper1 problems 3, 4

Page 9: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

PART 2FOLDING and UNFOLDING

In « military » perspective

Page 10: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Folding and unfolding cylindersin « military » perspective

Page 11: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

The technique

Paper1 problems 5

Page 12: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

The result

Paper1 problems 5

Page 13: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Folding and unfolding conesin « military » perspective

Page 14: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

The model

Paper2 problem 1

Page 15: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

PART 3The experimental process of discovery with technology

Two conjectures obtained with the model of unfolding a cone

and their proofs

Page 16: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Unfolding a cone onto half a disk

Paper2 problems 2

Page 17: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Formal proof

Page 18: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Evaluation of a limit of a ratio (between two angles)

Paper2 problem 3

Page 19: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Formal proof

Page 20: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

PART 4SURFACES z = f(x,y)

Two possible approaches

Page 21: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

With the « Graphs » application of TI N’Spire

Page 22: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Paper3 problem1

Page 23: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Paper3 problem 2

Page 24: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

With the « 3D Graphing » tool of TI N’Spire

Page 25: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

z = sin(x)+cos(y)

z = 0

Paper3 problem 3

Page 26: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

z = sin(x)+cos(y)

z = 0

Paper3 problem 4

Page 27: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

CONCLUSIONas a new title

Dynamic numbers for a dynamic approach of 3D analytic geometry

Page 28: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

z = sin(x)- k.cos(y)

Paper3 problem 5

Page 29: A Dynamic Approach of Analytic Geometry in 3D with TI N’Spire Enhancing an Experimental Process of Discovery Jean-Jacques Dahan jjdahan@wanadoo.fr IREM.

Dank

u [email protected]