mathcomesalive

- - - - -  - - - -3D Objects and Spline Curves- - - - - - - -


The display is a three dimensional perspective view of

the (geometrical) world, where the viewing plane is

the xy plane (z = 0, x horizontal, y vertical) and the

viewing point (the eye) is 20 units in front of this, on

the positive z axis.


- - - - - - - - - Three Dimensional Objects- - - - - - - - -


Select  "3D objects"

3D objects can be constructed from points by joining

pairs of points.  This is done by selecting 'Join own

points' and then clicking on the two end points that

you want to join, and then the next pair, and the next...


It is not easy to get numerically exact figures this way,

but you can have some fun trying.

Use  'Fetch' and select 'fetch6_octahedron joined' or

'fetch6_tetahedron joined' or 'fetch6_cube joined'.

These have been saved as points but with the linking

line segments saved as well.


Click 'Movements' to get the green menu box and do

things to the object chosen.

Set the +ve to -ve and do 'shift z'. This makes the

object move further away.

Rotate the object and see how the more distant edges

appear thinner than the nearer ones.  

Distort the object by moving one or more vertices

(movements are always in the xy plane, so dont be too

surprised at the results)


Now use Notepad to view one of the files, and by

following the structure try to create your own object

as a file (.txt), then fetch it into the program.


Investigate the examples on parallel lines to see if the

perspective view makes them "meet at infinity".



- - - - - - - - - - -Plane spline curves- - - - - - - - - - -.


Select "Spline curves"

Click 'Do own points' and then click on the xy plane

wherever you want a point.

Now click   'Do spline'.


The circle based spline is shown as a set of points, in

red.  The first and last points are not connected, but do

affect the direction of the curve as it leaves the

adjacent point.

'Show as line' replaces the points with a sequence of

line segments, and usually appears as a 'nice' curve.


You can now move any of the original points, shown in

black, and the spline curve is recalculated.


If you want a closed curve, select 'Set to closed'

before doing  'Do spline'.


Select 'Clear' to get rid of the current picture, and try

with three points , set to closed.  Surprised?


The movements of shift and stretch can be used on

your plane spline curve, but stick to the x and y

actions.

Try out the rotation actions, but these will be more

useful in 3D


'Reset' puts your points back in thier original

positions, with no spline curve.


- - - - - - - - - - -Stepping details- - - - - - - - - - -

The axes are marked in units, and the shift moves in

steps of 0.25 (a quarter) of a unit.

Stretching is done with a stretch factor of 1.2 for each

stretch, and the rotational angle step is pi/72, which

gives 36 steps for one right angle.


- - - - - - - - - - -3D spline curves- - - - - - - - - - -


To get some points in three dimensions set them up in

the xy plane, rotate around the x axis through a right

angle, move some of the points up or down, and then

rotate back through a right angle. You have now altered

the z values of the moved points.


You can do this, and then go back to the 'Do spline'

command, or you can do the spline on the plane curve

and do the rotation and moving afterwards.


Now rotate the figure around the x or the y axis, and

see how the further away points are shown in a fainter

color.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

There are a number of data sets set up which you can

get by doing 'Fetch'

Try 'bumpy hexagon'.  The points will be displayed, so

do 'Set to closed' and 'Do spline', and start a rotation

around the y axis.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - 3D spline surfaces- - - - - - - - - - -


Select  "Spline surfaces"

A surface is presumed to be covered by a grid of

quadrilaterals, in equal sized rows, and in which the

vertices are assumed to be the positions of the known

or measured values of the surface. The spline process

works its way along and then down the edges of the

quadrilaterals, creating estimates of the surface value

at intermediate points. Each quadrilateral is eventually

described by 64, mostly new, points, and an

approximation to the surface is obtained. The

procedure would be continued further in an

engineering environment.

There are 13 files of grids for you to investigate.

Select "Spline surfaces" and then "Fetch points"

When the points are displayed pick "Do spline"

Go to the movements menu to manipulate the picture -

make it bigger, rotate it etc.

Have a look at the point data in the files to figure out

what is going on.

You can also use "Do own points".  This will set up a

regular flat grid.

Use the "Change z values" to move some of the points

in or out and then do the spline.

3D_comes_alive

The help file together with illustrations

The two parts of the main menu

A cube, with depth (z axis) shown by thickness of line, fetched from one of the data files provided, and rotated using the Movements menu shown below:

These menus are selected, together, from the main menu, and provide a wide range of choices for movement and control of the image.

In particular, the rotate items cause the image to rotate continuously until the item is clicked again.

Two views of a 3D spline curve connecting ten points (the black ones. The curve can be viewed also as a line.

Splined bumpy hexagon

Two views of a roughly rectangular region, with bumps. This is constructed from a 5 by 6 grid of points. Closed or partially closed regions can also be “splined”.

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Math Comes Alive