Drawing 3D Lines and Boxes

[Eddie W. Shore]

Link to blog post: https://edspi31415.blogspot.com/2019/04/...boxes.html

Introduction

I was sent an email from Carissa. One of the questions was to explain the 3D features of LINE and TRIANGLE commands. Confession: I have never used to the 3D features of LINE and TRIANGLE before, time to learn something new. Here is what I learned.


The Command LINE

In 3D drawing, you can connect as many points as you want. In the most simple form, LINE takes three arguments:

* Point Definition
* Line Definition
* Rotation Matrix

Point Definition

You define points of three dimensions (x, y, z) and if you want, a color. The color attached to the point dominates any portion of the line that is nearest to the point. The point definition is a nested list.

Syntax: { {x, y, z, [color]}, {x, y, z, [color]}, {x, y, z, [color]}, ... }

Pay attention to the order you set your points. Each point will be tied to an index. For example, the first point in this list will have index 1, the second point in the list will have index 2, and so on. Knowing the index number will be needed for the Line Definition.

Line Definition

This is where you assign which lines connect to which points.

Syntax: { {index start, index end, [color], [alpha]}, {index start, index end, [color], [alpha]}, {index start, index end, [color], [alpha]}, ... }

Like the Point Definition, the Line Definition is a nested list. You can make as many points as you want.

Color and alpha in the List Definition overrides any color defined in the Point Definition list.

Rotation Matrix

This tells you the command how you want to rotate the matrix with the respect to the x, y, and z axis, respectively. The acceptable size for the matrix is 2 x 2 (for x and y only), 3 x 3 (x, y, and z axis) and 3 x 4 (I'm not sure what the fourth column is for). For this blog entry and in practice, I use the 3 x 3 rotation matrix.

In general:

Rx = [ [1, 0, 0],[0 cos a, -sin a],[0, sin a, cos a] ], rotation about the x axis at angle a
Ry = [ [cos b, 0, -sin b], [0, 1, 0], [sin b, 0, cos b ] ], rotation about the y axis at angle b
Rz = [ [cos c, -sin c, 0], [sin c, cos c, 0], [0, 0, 1] ], rotation about the z axis at angle c

Full Rotation matrix: r = Rx * Ry * Rz

You can create a program to calculate rotation matrix or copy the syntax to use in your drawing program, as shown here:

HP Program ROTMATRIX

Code:

EXPORT ROTMATRIX(a,b,c)
BEGIN
// rotate x axis
// rotate y axis
// rotate z axis
LOCAL x,y,z,r; 

x:=[[1,0,0],[0,COS(a),−SIN(a)],
[0,SIN(a),COS(a)]];
y:=[[COS(b),0,−SIN(b)],[0,1,0],
[SIN(b),0,COS(b)]];
z:=[[COS(c),−SIN(c),0],
[SIN(c),COS(c),0],[0,0,1]];
r:=x*y*z;

RETURN r;

END;

Drawing the Boxes

HP Prime Program: DRAWBOX

The program DRAWBOX draws a simple box.

Code:

EXPORT DRAWBOX()
BEGIN
// 3D line demo
// draw still box demo 
// 2019-04-23 EWS

// black background
RECT(0);

// colors - do this first
LOCAL c1:=#FFFF00h; // yellow
LOCAL c2:=#39FF14h; // neon green
LOCAL c3:=#FFFFFFh; // white


// points of the cube
LOCAL p:={{−3,0,0,c1},{0,−2,2,c1},
{3,0,0,c1},{0,2,−2,c2},
{−3,6,0,c1},{0,4,2,c1},
{3,6,0,c1},{0,8,−2,c2}};



// line definitions
// bottom
LOCAL d1:={{1,2},{2,3},
{3,4},{4,1}};
// top
LOCAL d2:={{5,6},{6,7},
{7,8},{8,5}};
// sides, override with white
LOCAL d3:={{1,5,c3},{2,6,c3},
{4,8,c3},{3,7,c3}};

// rotation matrix
LOCAL r:=[[1,0,0],[0,1,0],
[0,0,1]];

LINE(p,d1,r);
LINE(p,d2,r);
LINE(p,d3,r);

WAIT(0);

END;

HP Prime Program: DRAWBOX2

DRAWBOX2 takes three arguments, rotation of the x axis, rotation of the y axis, and rotation of the z axis. The arguments are entered in degrees, as the calculator is set to Degrees mode in the program.

Code:

EXPORT DRAWBOX2(a,b,c)
BEGIN
// 3D line demo
// draw 3D box
// 2019-04-23 EWS
// rotate the cube

// set degrees mode
HAngle:=1;

// rotation calculation
// rotate x axis
// rotate y axis
// rotate z axis
LOCAL x,y,z,r; 

x:=[[1,0,0],[0,COS(a),−SIN(a)],
[0,SIN(a),COS(a)]];
y:=[[COS(b),0,−SIN(b)],[0,1,0],
[SIN(b),0,COS(b)]];
z:=[[COS(c),−SIN(c),0],
[SIN(c),COS(c),0],[0,0,1]];
r:=x*y*z;

// black background
RECT(0);

// colors - do this first
LOCAL c1:=#FFFF00h; // yellow
LOCAL c2:=#39FF14h; // neon green
LOCAL c3:=#FFFFFFh; // white


// points of the cube
LOCAL p:={{−3,0,0,c1},{0,−2,2,c1},
{3,0,0,c1},{0,2,−2,c2},
{−3,6,0,c1},{0,4,2,c1},
{3,6,0,c1},{0,8,−2,c2}};



// line definitions
// bottom
LOCAL d1:={{1,2},{2,3},
{3,4},{4,1}};
// top
LOCAL d2:={{5,6},{6,7},
{7,8},{8,5}};
// sides, override with white
LOCAL d3:={{1,5,c3},{2,6,c3},
{4,8,c3},{3,7,c3}};

// rotation matrix is already
// defined

LINE(p,d1,r);
LINE(p,d2,r);
LINE(p,d3,r);

WAIT(0);

END;

3D Triangles

The format for TRIANGLE is similar except the definition list has three points to make up the triangle instead of two. Format: {x, y, z, [ c ]}

Code:

Code:

EXPORT TEST6241()
BEGIN
// test 3D triangle
RECT_P();

//TRIANGLE({0,0,#0h,0},
//{2,2,#0000FFh,39},
//{1,5,#FF0000h,−2});

LOCAL c:=#400080h;
LOCAL p:={{0,0,0},{−10,−8,2},
{−6,4,3},{0,0,0}};
LOCAL d:={{1,2,3,c},{1,2,4,c},
{1,3,4,c},{2,3,4,c}};
local rotmat= [[1, .5, 0], [.5, 1, .5], [0, .5, 1]];   // Initial rotation matrix. No rotation but translate to middle of screen 
TRIANGLE(p,d,rotmat);

WAIT(0);
END;

Hope this helps and all of our drawing capabilities on the HP Prime are expanded. Carissa, thank you for your email, much appreciated and I learned a great new skill. All the best!

[Han]

Quote:and 3 x 4 (I'm not sure what the fourth column is for).

It designates the distance of the the "viewing screen" from the origin:

https://hpmuseum.org/forum/thread-5122.html

This final column is what gives 3D perspective.

[Eddie W. Shore]

Thank you Han!

Eddie