function GAdemo% GAdemo: a short demonstrations of the basics of GA%%See also gable.% GABLE, Copyright (c) 1999, University of Amsterdam% Copying, use and development for non-commercial purposes permitted.%          All rights for commercial use reserved; for more information%          contact Leo Dorst (leo@wins.uva.nl).%%          This software is unsupported.tryclcdisp('Welcome to the GABLE demonstration!');disp(' ');disp('This script demonstrates some of the basic features of GABLE.');disp('At any prompt, you may hit return to continue the demo or type');disp('a Matlab command.');disp(' ');disp('This brief demonstration is intended to introduce you to the');disp('the basic geometric elements of our geometric algebra (vectors,');disp('bivectors, and trivector) and introduce the three basic products:');disp('the outer product, the inner product, and the geometric product.');disp('For a discussion of what these products are useful for, see our');disp('tutorial.');disp(' ');GAprompt;disp(' ');disp('GABLE is an implementation of a Geometric Algebra (Cl3,0) with');disp('graphical representations of the geometrical objects.  The defining');disp('objects of a Geometric Algebra are a basis of vectors.  In GABLE,');disp('we use a vector basis of {e1,e2,e3}');disp(' ');GAprompt;disp(' ');disp('We will now draw e1, e2, and e3, in the graphics window.');disp('As we draw things, we will list the corresponding graphics');disp('commands on lines starting with '' >> ''');disp(' ');disp('In our software, basic objects are drawn with the ''draw'' command,');disp('which has an optional color argument.');pause(5);disp(' >> clf;');clf;disp(' >> draw(e1,''r'');');draw(e1,'r');drawnowdisp(' >> draw(e2,''m'');');draw(e2,'m');drawnowdisp(' >> draw(e3,''b'');');draw(e3,'b');drawnow;disp(' ');pause(1);disp('In the graphics window, you should see three arrows.  The red one');disp('represents e1, the magenta one e2, and the blue one e3.  Matlab');disp('automatically rescales the axes as needed.');disp(' ');GAprompt;disp(' ');clcdisp('There are three products in a Geometric Algebra, the first of which');disp('we will look at being the Outer product.  In GABLE, we use');disp('the circumflx symbol (^) to take the outer product of two objects.');disp('The outer product of two vectors is a bivector, which can be thought');disp('of as a directed area element in the plane spanned by the two bivectors.');disp(' ');GAprompt;disp(' ');disp('As an example of a bivector, we will draw e1^e2.');disp(' >> draw(e1^e2);');draw(e1^e2);drawnowdisp(' ');disp('We have drawn a circle to represent this bivector.  Note that');disp('the bivector drawn for e1^e2 lies in the plane defined by');disp('e1 and e2');disp(' ');GAprompt;disp(' ');disp('To give a better feeling of the 3D relationships of the objects,');disp('we have implemented a routine called GAorbit that rotates the');disp('scene.  We will run it now to allow you a better view of the');disp('three vectors and the bivector.');pause(4);disp(' >> GAorbit(-180)');GAorbit(-180)disp(' ');disp('At prompts, you may find it useful to type GAorbit to visualize the');disp('scene more clearly.  Try that now.');disp(' ');GAprompt;disp(' ');disp('The outer product of two objects yields a higher dimensional');disp('object only if the two objects are linearly independent.  Thus,');disp('the outer product e1^e1 is zero.  Type ''e1^e1'' in now and see');disp('for yourself.');disp(' ');GAprompt;disp(' ');disp(' ');disp('In addition to vectors and bivectors, there are trivectors,');disp('which are formed as the outer product of three independent vectors.');disp('We will now draw the trivector e1^e2^e3 (which has a special name,');disp('''I3'') on the screen...');disp(' >> draw(I3);');draw(I3);drawnowpause(2);disp(' ');disp('We have drawn the trivector as a line drawing of a sphere, although it');disp('should be thought of as a solid volume element.  In our space, the');disp('trivector is the highest dimensional object, and the outer product');disp('of the trivector with any other element in our space gives us zero,');disp('since the two objects must be linearly dependent.');disp(' ');GAprompt;disp(' ');disp('Visualizing bivectors as circles and trivectors as spheres is not');disp('the usual way to visualize them.  See the tutorial for another method');disp('of visulazation.');disp(' ');GAprompt;clc;disp('The second product in a Geometric Algebra is the inner product,');disp('which is similar to the dot product on vectors.  However, on');disp('objects of different dimensions, it finds the object lying in');disp('the higher dimensional space that is perpendicular to the lower');disp('dimensional object.');disp(' ');GAprompt;disp(' ');disp(' ');disp('We implemented the inner product as a Matlab routine called "inner"');disp('As an example of the inner product, we draw the vector e1+e3 and the');disp('bivector e1^e2.');disp(' >> clf;');clf;pause(1);disp(' >> draw(e1+e3);');draw(e1+e3);drawnowdisp(' >> draw(e1^e2);');draw(e1^e2);drawnowdisp(' ');disp('(In Matlab, the command ''clf'' clears the display.)');disp(' ');GAprompt;disp(' ');disp('Next, we will draw in red the inner product of these two ');disp('objects, inner(e1+e3,e1^e2), which we rotate for a better view...');pause(3);disp(' >> draw(inner(e1+e3,e1^e2),''r'');');draw(inner(e1+e3,e1^e2),'r');disp(' >> GAorbit(-180)');pause(1);GAorbit(-180)disp(' ');GAprompt;disp(' ');clcdisp('The third and most important product of a Geometric Algebra is the');disp('Geometric Product.  Unfortunately, the geometric product is not');disp('easily visualized.  However, one nice property it has that the other');disp('two products do not have is an inverse.  The inverse allows you');disp('to divide by vectors, bivectors, etc.');disp(' ');GAprompt;disp(' ');disp('Rather than visualize an object created by the geometric product, it');disp('is better to think of it as creating an operator, and to visualize');disp('this operator being applied to an object.  For example, if you have');disp('a vector b and want to rotate it to vector a, you are looking for');disp('an operator R such that b = Ra.  This suggests that R = b/a, which');disp('we can compute with the geometric product.');disp(' ');GAprompt;disp(' ');disp('So try this out.  We''ll clear the screen and create the vectors for you:');disp(' >> clf;');clf;disp(' >> a = (e1+e2)/sqrt(2);');a = (e1+e2)/sqrt(2);disp(' >> b = e2;');b = e2;pause(3);disp('Now at the prompt create R, draw a, and draw b.');disp('To get a 2D perspective on your picture, type GAview([0 90]) after drawing');disp(' ');GAprompt;disp(' ');disp('Now draw draw R*a in red (give a second argument of ''r'' to draw to get red).');disp(' ');GAprompt;disp(' ');disp('What you see is that the red vector (R*a) overwrote the blue b vector');disp('as desired, since R is suppose to rotate a into b');disp('Try rotating a second and third time by drawing R*R*a and R*R*R*a.');disp(' ');GAprompt;disp(' ');disp('This method of representation of rotations has some draw-backs that');disp('we won''t discuss here; see the tutorial for a discussion of those ');disp('draw-backs and a better representation of rotations.');disp(' ');disp('This is the end of our brief introduction to GABLE.');disp('For a further discussion on Geometric Algebra, see our tutorial,');disp('which investigates other visualizations of bivectors and trivectors,');disp('and shows how to compute useful geometric calculations using');disp('the three products described in this demonstration.');catch ; end