Tips for the Plane Graphic Calculator
...and Teach the basics of a geometric vectorspace
Plane Graphic Calculator from J.E.Lillge
One of the wonders of planegraph is, that it is a powerful
2D numerical calculator
too. Click on "Detach" and on "Details". Enter into the f-field
of planegraph any numbers and vectors and 2x2-matrices and
nearly every operation you can think of - and hit the Enter-key on
your keyboard. The result is shown in the f-field of "Details".
And another wonder of planegraph is, that you can calculate in
too, just like my
"Forget C, it's all R2, it's all 2D- calculator".
So Planegraph is the v e r y f i r s t, who has this property.
i * (3 , 4)
or: ( - 5 + 4 * i) * ( 4.1 , 3 )
And You can calculate sin (2,5) or (3+4i)dot(6+9i),..
When You enter into f a formula or a term of a, You can
move a on the plane, and the dependent f-vector has to follow,
both displayed as locations, positions, as arrows fixed to the origin.
You want to display a function f=f(a) as
vector field ?
"Here is an example showing how you can enter your
vector field for (a^2+sin a) into the calculator:
g = a+(a^2+sin a)(t>0)
(You will probably want to make sure that the
"Line" checkbox is not checked.)
As you can see, the vector is drawn from the
tip of a to the tip of a+f, where:
f = (a^2+sin a)
The general form for displaying a vector field in the
g = a+(f)(t>0)
where (f) is replaced by any parenthesized
vector function of a."
J.E.Lillge wrote to me. So now we can move the vector a
on the plane like a probe investigating this field,
and the vector f will follow.
This feature gives the beautiful possibility to
Teach the basics of a geometric vectorspace
Use the keyboard and let the mouse to Your pupil.
Click on Detach and on Detail, click on the three
boxes Plot, Spin, Line until they are all blank.
Change in the box function to: user-defined.
Enter a vector 1+2i or (1,2) into m by typing
a + ((1+2i)) (t>0) (Enter)
a denotes the tail of an arrow and (t>0) draws it
Your pupil will see:
an arrow - with a very sharp tip. He can grab it by the tail,
the point a and move it around.
A vector is geomtrically an arrow, a straight line-segment
with a direction.The direction relates to the environment
and/or the other arrows. By translating the arrow, you don't
Enter a second vector (1.5, - 0.7) into n:
b + ((1.5, - 0.7)) (t>0).
Into f You'll always enter Your calculation,
here "a+B" or "a-b":
c + ((1+2i) + (1.5, - 0.7)) (t>0).
So the pupil can grasp the addition of arrows. Of course
You will show him some different vectors too.
In geometry You have learned, how to magnify/minimize or
resize a straight line-segment: attach a second line at an
angle betweeen 30 to 60 degrees to it, mark with compasses
a unit and any length, let's say three times the unit, on it.
(With compasses You can devide the unit in the middle,
the parts again in their middle, and by repeating this, you
can get the part behind the decimal point of any real number,
in a binary form).
Join the unit-mark on this "scale" with the end of the
straight line-segment and draw a parallel through the
3. - mark on the "scale". Where this meets the prolonged
line-segment is the end of the resized line-segment.
Enter into n:
b + ((3.5 * (1,2)) (t>0)
b displays the 3.5 - fold of a, it's a multiplication of an
arrow by a number, called: real scalar multiplication (r.s.m.).
Addition and r.s.m. together makes the set of directed
line-segments or arrows on a plane to a geometric vectorspace.
You can calculate and display linear combinations of arrows.
Enter into n:
b + ((1.5, - 0.7)) (t>0)
Enter into f:
c + ((2*(1+2i) - 0.5 *(1.5, - 0.7)) (t>0).
It's how Wessel started in the old-greek geometry style,
without the underlying grid, without a coordinate system,
an origin, a scale or a unit, without a pricipal or basic
direction. And it's good not only for 2D, but for 3D too,
and not only on a paper. Do it in reality with the pupil:
Go three paces north, turnleft and go another seven paces,
six steps up the stairs ,..
...the geometric Bombelli vectorspace
We add a multiplication (2D only).
In effect it's a rotation (addition of angles) and a
stretching/shrinking (multiplication of length).
It's only 2D, because on a plane all axes of rotation
are perpendicular (or normal) to the plane - thus
only parallel axes are used.
(To my best knowledge the only known representation of
a 2D - vectorspace with multiplication like this
is a plane with it's arrows.)
It's called a multiplication, because it's one extension
of the normal multiplication of (real) numbers, just
like the addition of arrows contains the addition
of length on one line, id est numbers.
We didn't represent numbers on the plane up to now, but
we did use them as factors in the r.s.m.
The common geometric representation of a quantity, a
number is a straight and sometimes a circle line-segment.
And a product is an area, a rectangle , but not a
line-segment. So You can't multiply two line-segments,
unless you introduce a unit or a scale.
Enter into m:
a + ((1)) (t>0)
and into n:
b + ((1.5)) (t>0)
and into f:
c + ((dx*(1)) * (dx *(1.5)))(t>0)
You see how the product "a times b" changes, when You
change the length of the unit d by moving d horizontal.
The multiplication of two straight line-segments of
length in terms of a unit is done geometrically somehow
similar to the r.s.m.:draw two line-segments at an angle
to each other, mark on one the unit and connect this
mark to the tip of the other. Draw a parallel through
the tip of the first and this meets the prolonged second
line segment in the tip of the resulting line-segment.
You can not multiply directions.
In the long math-lessons i often wondered, that multiplication
of vectors is represented by rotations, but they never told,
what it is, what they rotated.
Let's start with difference of direction, with division
(that's how Hamilton started with Quaternions).
From north-east to north you get the same difference of
direction, as from -let's say - south-west to south,
1/8 th of a complete circle - and these rotations are
orientated both anticlockwise or with left-turn. And
we will stay with this orientation left turning, so a
difference from south to southwest turning clockwise is
not admitted here. Of course You can rotate from south
to south-west anticlockwise, but the angle is nearly
a full circle (the conjugate angle for south-west to
Difference of direction - that's called an angle.
Oriented (here:left-turn) difference of direction -
that's an oriented angle.
So now we disentangle arrow b from the environment and
the direction of the other vectors.
Enter into m:
a + ((1.5,1)) (t>0)
and into n the arrow (2,1.4) or 2+i*1.4 by:
b + d/|d| *((2,1.4))(t>0) + d/|d| *( - (2,1.4))(t>1)
+ d/|d| *((cos t, sin t))(t>6.28)
and into Parametric value for t:
t from 0 to 6.89
as (6.89=2*pi + 35/360 * 2*pi)
and into f a zero.
If You want to display the unit enter into f:
c + (1)(t>0),
As You see, the reference-direction and the arc associated
with b has a radius of one unit.
Move b by its tail and by changing d on the plane.
An arrow with an arc of zero degrees (t from 0 to 6.28) is
a line segment, a length without direction.
Finally we can multiply a * b, a rotation of the direction
of a by the arc of b and with orientation left-turn and the
length of both arrows multiplied with regard to the unit:
Enter into f:
c + ((1.5,1) * (2, 1.4)) (t>0).
When You proceed to multiply or divide two rotations, the best
is to open up planegraph a second time, so You get four more
indepent handles (variables) and one window extra to display
That you can't multiply two directions is not specific for
2D or this multiplication. You can't add house-number 3 to
house-number 7, or add two dates.They are called ordinal-numbers.
By walking from house-number 3 a kardinal-number of 4 houses on
you get to house-number 7.
....the analytical representation of arrows as vectors
that's, what You have done all the time by entering numbers and vectors.
They relate to the underlying grid. So You introduce an origin,
a pricipal or basic direction - in short a coordinate-system.
You will notice differences between an algebraic and the geometric
representation of a vector-space. You can check on the
definitions, for example: For every arrow exists an (opposite)
arrow and they add up to zero, a point without direction.
And You can proceed to the projection of an arrow onto the axes,
the dot-product and the Euclid-vector-space.
May be You have noticed, that the axes of the rotations
are not fixed to a point on the plane: a ship can take a turn left
around a fire-house, the axis or the mast is moving and the ship
and the direction of travel rotates.