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I'd like to programmatically draw a shape like this where there is an underlying spiral and equally spaced objects along it, placed tangent to the spiral as shown in this sketch:
I found an example of how to determine equally spaced points along the spiral here and am now trying to place hemispheres along the spiral. However, I'm not sure how to calculate the angle the shape needs to be rotated.
This is what I have so far (viewable here):
var totalSegments = 235,hw = 320,hh = 240,segments;
var len = 15;
points = [];
function setup(){
createCanvas(640,480);
smooth();
colorMode(HSB,255,100,100);
stroke(0);
noFill();
//println("move cursor vertically");
}
function draw(){
background(0);
translate(hw,hh);
segments = floor(totalSegments);
points = getTheodorus(segments,len);
angles = getAngles(segments, len);
for(var i = 0 ; i < segments ; i++){
let c = color('blue');
fill(c);
noStroke();
// draw shape
if(i % 2){
// console.log(i, ' ', angles[i]);
// try rotating around the object's center
push();
// translate(points[i].x, points[i].y)
rotate(PI/angles[i]);
arc(points[i].x, points[i].y, len*3, len*3, 0, 0 + PI);
pop();
}
// draw spiral
strokeWeight(20);
stroke(0,0,100,(20+i/segments));
if(i > 0) line(points[i].x,points[i].y,points[i-1].x,points[i-1].y);
}
}
function getAngles(segment, len){
let angles = [];
let radius = 0;
let angle = 0;
for(var i =0; i < segments; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
angles[i] = angle;
}
return angles;
}
function getTheodorus(segments,len){
var result = [];
var radius = 0;
var angle = 0;
for(var i = 0 ; i < segments ; i++){
radius = sqrt(i+1);
angle += asin(1/radius);
result[i] = new p5.Vector(cos(angle) * radius*len,sin(angle) * radius*len);
}
return result;
}
Note that your drawing shows Archimedean spiral while link refers to Theodorus one.
Archimedean spiral is described by equation in polar coordinates (rho-theta)
r = a + b * Theta
where a is initial angle, b is scale value (describes distance between arms), r is radius.
And angle Theta + Pi/2 describes normal to spiral in point at parameter Theta
If you need an approximation to divide spiral into (almost) equal segments - use Clackson formula (example here)
theta = 2 * Pi * Sqrt(2 * s / b)
for arc length s
I want bevel the sides of a rectangle in order to do so i want to draw a circle from the start point to the end point as shown in the image
while drawing a complete circle this is how i do it
float angle = 2.0f * M_PI * i / iSegments;
// vertex data
float x, y, z ,tx,ty ,tz;
x = cos(angle) * 50.0;
y = sin(angle) * 50.0;
z = 0.0;
How do we calculate the vertices for the circle in the given case ?
The center of the circle is radius away from the corner point, both in x and in y directions. In your example the radius is 30. To draw an arc of 90° (π/2 radians), you can start with the first angle and divide π/2 into as many segments as you want. The first angle depends on the which corner of the rectangle is treated. In the example of the upper right corner, the starting angle would be 0°.
Here is some code to illustrate the concept:
// draw a circular bevel to a rectangle, given are
// p_x, p_y: the coordinates of the corner, e.g. 50, 50
// rad: the radius of the bevel, e.g. 30
// dir_x: the direction to indicate whether the center of
// the circle lies lef (dir_y=-1) or right (dir_x=+1) of the corner
// dir_y: the direction to indicate whether the center of
// the circle lies lower (dir_y=-1) or higher (dir_y=+1) than the corner
void draw_circular_bevel (float p_x, float p_y, float rad, int dir_x, int dir_y)
{
float c_x, c_y; // the center of the circle
float start_angle; // the angle where to start the arc
c_x = p_x + rad * dir_x;
c_y = p_y + rad * dir_y;
if (dir_x == 1 and dir_y == 1)
start_angle = 0.0;
else if (dir_x == 1 and dir_y == -1)
start_angle = - M_PI * 0.5f;
else if (dir_x == -1 and dir_y == 1)
start_angle = M_PI * 0.5f;
else if (dir_x == -1 and dir_y == -1)
start_angle = - 2.0f * M_PI;
for (int i=0; i <= iSegments; ++i) {
float x, y;
float angle = start_angle + 0.5f * M_PI * i / (float)iSegments;
x = c_x + cos(angle) * rad;
y = c_y + sin(angle) * rad;
// here call code to draw a point or a segment at position x,y
}
}
I'm developing a Processing sketch that, given a certain angle, draws a dot at the edge of a rhombus.
I know the width of the rhombus, and its position, but I'm not sure how to calculate the x-y coordinates of a dot resting at its edge.
Are there any elegant solutions for this problem? Any help in pseudocode would be welcomed.
Let's square side length is A, half-length is H = A/2. Angle Theta. Intersection point P.
All coordinates are relative to the square center.
Rotate square by -Pi/4, angle Alpha = Theta - Pi/4
if Alpha lies in range -Pi/4..Pi/4, then intersection point P' = (H, H*Tan(Alpha))
if Alpha lies in range Pi/4..3*Pi/4, then P' = (H*Cotangent(Alpha), H)
if Alpha lies in range 3*Pi/4..5*Pi/4, then P' = (-H, -H*Tan(Alpha))
if Alpha lies in range 5*Pi/4..7*Pi/4, then P' = (-H*Cotangent(Alpha), -H)
Then rotate point P' back by Pi/4:
S = Sqrt(2)/2
P.X = S * (P'.X - P'.Y)
P.Y = S * (P'.X + P'.Y)
Example (data like your sketch):
A = 200, Theta = 5*Pi/12
H = 200/2 = 100, Alpha =Theta-Pi/4 = Pi/6
P'.X = H = 100
P'.Y = H * Tan(Alpha) = 100 * Tan(Pi/6) ~= 57.7
S = 0.707
P.X = 0.707 * (100 - 57.7) = 30
P.Y = 0.707 * (100 + 57.7) = 111
Based on your image, you want to find the intersection of two equations, that of the line at angle θ, and that of the side of the square with which it intersects.
Assuming the size of your square is n, the equation of the square is y=±(n*(√2/2))±x (by Pythagoras' theorem). The equation for the side you intersect in your image is y=n*(√2/2)-x.
The equation of the radial line can be calculated using trigonometry to be y=tan(θ)*x, with θ expressed in radians.
You can then solve this as a simultaneous equation to determine the intersection. Please note that it will intersect with both sides of the square (both above and below), so if you only want the one you will have to choose the equation for the correct side of the square. Also guard against the case where θ is π/2, as tan(π/2) is undefined. You can easily work out that case, as x=0 and so it will always intersect at y=±(n*(√2/2)).
In your example, the intersection occurs when x*(1+tan(θ))=n*(√n/n), or x=(n*(√n/n))/(1+tan(θ)). You can calculate that, plug it back into y and that is your (x,y) intersection.
Imagine a circle with a larger radius that will intersect your rhombus at the points you want. One way to draw at that location is to use a nested coordinate system that you translate and rotate. All you need to know is the radius and the angle.
Here's a very basic example:
float angle = radians(-80.31);
float radius = 128;
float centerX,centerY;
void setup(){
size(320,320);
noFill();
rectMode(CENTER);
centerX = width * 0.5;
centerY = height * 0.5;
}
void draw(){
background(255);
noFill();
//small circle
strokeWeight(1);
stroke(95,105,120);
ellipse(centerX,centerY,210,210);
rhombus(centerX,centerY,210);
//large circle
strokeWeight(3);
stroke(95,105,120);
ellipse(centerX,centerY,radius * 2,radius * 2);
//line at angle
pushMatrix();
translate(centerX,centerY);
rotate(angle);
stroke(162,42,32);
line(0,0,radius,0);
popMatrix();
//debug
fill(0);
text("angle: " + degrees(angle),10,15);
}
void rhombus(float x,float y,float size){
pushMatrix();
translate(x,y);
rotate(radians(45));
rect(0,0,size,size);
popMatrix();
}
void mouseDragged(){
angle = atan2(centerY-mouseY,centerX-mouseX)+PI;
}
You can try a demo here(you can drag the mouse to change the angle):
var angle;
var radius = 128;
var centerX,centerY;
function setup(){
createCanvas(320,320);
noFill();
rectMode(CENTER);
angle = radians(-80.31);
centerX = width * 0.5;
centerY = height * 0.5;
}
function draw(){
background(255);
noFill();
//small circle
strokeWeight(1);
stroke(95,105,120);
ellipse(centerX,centerY,210,210);
rhombus(centerX,centerY,210);
//large circle
strokeWeight(3);
stroke(95,105,120);
ellipse(centerX,centerY,radius * 2,radius * 2);
//line at angle
push();
translate(centerX,centerY);
rotate(angle);
stroke(162,42,32);
line(0,0,radius,0);
pop();
//debug
fill(0);
noStroke();
text("angle: " + degrees(angle),10,15);
}
function rhombus(x,y,size){
push();
translate(x,y);
rotate(radians(45));
rect(0,0,size,size);
pop();
}
function mouseDragged(){
angle = atan2(centerY-mouseY,centerX-mouseX)+PI;
}
<script src="https://cdnjs.cloudflare.com/ajax/libs/p5.js/0.5.0/p5.min.js"></script>
If you want to calculate the position, you can use the polar to cartesian coordinate conversion formula:
x = cos(angle) * radius
y = sin(angle) * radius
Here's an example using that. Note that drawing is done from the centre, therefore the centre coordinates are added to the above:
float angle = radians(-80.31);
float radius = 128;
float centerX,centerY;
void setup(){
size(320,320);
noFill();
rectMode(CENTER);
centerX = width * 0.5;
centerY = height * 0.5;
}
void draw(){
background(255);
noFill();
//small circle
strokeWeight(1);
stroke(95,105,120);
ellipse(centerX,centerY,210,210);
rhombus(centerX,centerY,210);
//large circle
strokeWeight(3);
stroke(95,105,120);
ellipse(centerX,centerY,radius * 2,radius * 2);
//line at angle
float x = centerX+(cos(angle) * radius);
float y = centerX+(sin(angle) * radius);
stroke(162,42,32);
line(centerX,centerY,x,y);
//debug
fill(0);
text("angle: " + degrees(angle),10,15);
}
void rhombus(float x,float y,float size){
pushMatrix();
translate(x,y);
rotate(radians(45));
rect(0,0,size,size);
popMatrix();
}
void mouseDragged(){
angle = atan2(centerY-mouseY,centerX-mouseX)+PI;
}
Another option would be using transformation matrices
i'm trying to code correct 2D affine texture mapping in GLSL.
Explanation:
...NONE of this images is correct for my purposes. Right (labeled Correct) has perspective correction which i do not want. So this: Getting to know the Q texture coordinate solution (without further improvements) is not what I'm looking for.
I'd like to simply "stretch" texture inside quadrilateral, something like this:
but composed from two triangles. Any advice (GLSL) please?
This works well as long as you have a trapezoid, and its parallel edges are aligned with one of the local axes. I recommend playing around with my Unity package.
GLSL:
varying vec2 shiftedPosition, width_height;
#ifdef VERTEX
void main() {
gl_Position = gl_ModelViewProjectionMatrix * gl_Vertex;
shiftedPosition = gl_MultiTexCoord0.xy; // left and bottom edges zeroed.
width_height = gl_MultiTexCoord1.xy;
}
#endif
#ifdef FRAGMENT
uniform sampler2D _MainTex;
void main() {
gl_FragColor = texture2D(_MainTex, shiftedPosition / width_height);
}
#endif
C#:
// Zero out the left and bottom edges,
// leaving a right trapezoid with two sides on the axes and a vertex at the origin.
var shiftedPositions = new Vector2[] {
Vector2.zero,
new Vector2(0, vertices[1].y - vertices[0].y),
new Vector2(vertices[2].x - vertices[1].x, vertices[2].y - vertices[3].y),
new Vector2(vertices[3].x - vertices[0].x, 0)
};
mesh.uv = shiftedPositions;
var widths_heights = new Vector2[4];
widths_heights[0].x = widths_heights[3].x = shiftedPositions[3].x;
widths_heights[1].x = widths_heights[2].x = shiftedPositions[2].x;
widths_heights[0].y = widths_heights[1].y = shiftedPositions[1].y;
widths_heights[2].y = widths_heights[3].y = shiftedPositions[2].y;
mesh.uv2 = widths_heights;
I recently managed to come up with a generic solution to this problem for any type of quadrilateral. The calculations and GLSL maybe of help. There's a working demo in java (that runs on Android), but is compact and readable and should be easily portable to unity or iOS: http://www.bitlush.com/posts/arbitrary-quadrilaterals-in-opengl-es-2-0
In case anyone's still interested, here's a C# implementation that takes a quad defined by the clockwise screen verts (x0,y0) (x1,y1) ... (x3,y3), an arbitrary pixel at (x,y) and calculates the u and v of that pixel. It was originally written to CPU-render an arbitrary quad to a texture, but it's easy enough to split the algorithm across CPU, Vertex and Pixel shaders; I've commented accordingly in the code.
float Ax, Bx, Cx, Dx, Ay, By, Cy, Dy, A, B, C;
//These are all uniforms for a given quad. Calculate on CPU.
Ax = (x3 - x0) - (x2 - x1);
Bx = (x0 - x1);
Cx = (x2 - x1);
Dx = x1;
Ay = (y3 - y0) - (y2 - y1);
By = (y0 - y1);
Cy = (y2 - y1);
Dy = y1;
float ByCx_plus_AyDx_minus_BxCy_minus_AxDy = (By * Cx) + (Ay * Dx) - (Bx * Cy) - (Ax * Dy);
float ByDx_minus_BxDy = (By * Dx) - (Bx * Dy);
A = (Ay*Cx)-(Ax*Cy);
//These must be calculated per-vertex, and passed through as interpolated values to the pixel-shader
B = (Ax * y) + ByCx_plus_AyDx_minus_BxCy_minus_AxDy - (Ay * x);
C = (Bx * y) + ByDx_minus_BxDy - (By * x);
//These must be calculated per-pixel using the interpolated B, C and x from the vertex shader along with some of the other uniforms.
u = ((-B) - Mathf.Sqrt((B*B-(4.0f*A*C))))/(A*2.0f);
v = (x - (u * Cx) - Dx)/((u*Ax)+Bx);
Tessellation solves this problem. Subdividing quad vertex adds hints to interpolate pixels.
Check out this link.
https://www.youtube.com/watch?v=8TleepxIORU&feature=youtu.be
I had similar question ( https://gamedev.stackexchange.com/questions/174857/mapping-a-texture-to-a-2d-quadrilateral/174871 ) , and at gamedev they suggested using imaginary Z coord, which I calculate using the following C code, which appears to be working in general case (not just trapezoids):
//usual euclidean distance
float distance(int ax, int ay, int bx, int by) {
int x = ax-bx;
int y = ay-by;
return sqrtf((float)(x*x + y*y));
}
void gfx_quad(gfx_t *dst //destination texture, we are rendering into
,gfx_t *src //source texture
,int *quad // quadrilateral vertices
)
{
int *v = quad; //quad vertices
float z = 20.0;
float top = distance(v[0],v[1],v[2],v[3]); //top
float bot = distance(v[4],v[5],v[6],v[7]); //bottom
float lft = distance(v[0],v[1],v[4],v[5]); //left
float rgt = distance(v[2],v[3],v[6],v[7]); //right
// By default all vertices lie on the screen plane
float az = 1.0;
float bz = 1.0;
float cz = 1.0;
float dz = 1.0;
// Move Z from screen, if based on distance ratios.
if (top<bot) {
az *= top/bot;
bz *= top/bot;
} else {
cz *= bot/top;
dz *= bot/top;
}
if (lft<rgt) {
az *= lft/rgt;
cz *= lft/rgt;
} else {
bz *= rgt/lft;
dz *= rgt/lft;
}
// draw our quad as two textured triangles
gfx_textured(dst, src
, v[0],v[1],az, v[2],v[3],bz, v[4],v[5],cz
, 0.0,0.0, 1.0,0.0, 0.0,1.0);
gfx_textured(dst, src
, v[2],v[3],bz, v[4],v[5],cz, v[6],v[7],dz
, 1.0,0.0, 0.0,1.0, 1.0,1.0);
}
I'm doing it in software to scale and rotate 2d sprites, and for OpenGL 3d app you will need to do it in pixel/fragment shader, unless you will be able to map these imaginary az,bz,cz,dz into your actual 3d space and use the usual pipeline. DMGregory gave exact code for OpenGL shaders: https://gamedev.stackexchange.com/questions/148082/how-can-i-fix-zig-zagging-uv-mapping-artifacts-on-a-generated-mesh-that-tapers
I came up with this issue as I was trying to implement a homography warping in OpenGL. Some of the solutions that I found relied on a notion of depth, but this was not feasible in my case since I am working on 2D coordinates.
I based my solution on this article, and it seems to work for all cases that I could try. I am leaving it here in case it is useful for someone else as I could not find something similar. The solution makes the following assumptions:
The vertex coordinates are the 4 points of a quad in Lower Right, Upper Right, Upper Left, Lower Left order.
The coordinates are given in OpenGL's reference system (range [-1, 1], with origin at bottom left corner).
std::vector<cv::Point2f> points;
// Convert points to homogeneous coordinates to simplify the problem.
Eigen::Vector3f p0(points[0].x, points[0].y, 1);
Eigen::Vector3f p1(points[1].x, points[1].y, 1);
Eigen::Vector3f p2(points[2].x, points[2].y, 1);
Eigen::Vector3f p3(points[3].x, points[3].y, 1);
// Compute the intersection point between the lines described by opposite vertices using cross products. Normalization is only required at the end.
// See https://leimao.github.io/blog/2D-Line-Mathematics-Homogeneous-Coordinates/ for a quick summary of this approach.
auto line1 = p2.cross(p0);
auto line2 = p3.cross(p1);
auto intersection = line1.cross(line2);
intersection = intersection / intersection(2);
// Compute distance to each point.
for (const auto &pt : points) {
auto distance = std::sqrt(std::pow(pt.x - intersection(0), 2) +
std::pow(pt.y - intersection(1), 2));
distances.push_back(distance);
}
// Assumes same order as above.
std::vector<cv::Point2f> texture_coords_unnormalized = {
{1.0f, 1.0f},
{1.0f, 0.0f},
{0.0f, 0.0f},
{0.0f, 1.0f}
};
std::vector<float> texture_coords;
for (int i = 0; i < texture_coords_unnormalized.size(); ++i) {
float u_i = texture_coords_unnormalized[i].x;
float v_i = texture_coords_unnormalized[i].y;
float d_i = distances.at(i);
float d_i_2 = distances.at((i + 2) % 4);
float scale = (d_i + d_i_2) / d_i_2;
texture_coords.push_back(u_i*scale);
texture_coords.push_back(v_i*scale);
texture_coords.push_back(scale);
}
Pass the texture coordinates to your shader (use vec3). Then:
gl_FragColor = vec4(texture2D(textureSampler, textureCoords.xy/textureCoords.z).rgb, 1.0);
thanks for answers, but after experimenting i found a solution.
two triangles on the left has uv (strq) according this and two triangles on the right are modifed version of this perspective correction.
Numbers and shader:
tri1 = [Vec2(-0.5, -1), Vec2(0.5, -1), Vec2(1, 1)]
tri2 = [Vec2(-0.5, -1), Vec2(1, 1), Vec2(-1, 1)]
d1 = length of top edge = 2
d2 = length of bottom edge = 1
tri1_uv = [Vec4(0, 0, 0, d2 / d1), Vec4(d2 / d1, 0, 0, d2 / d1), Vec4(1, 1, 0, 1)]
tri2_uv = [Vec4(0, 0, 0, d2 / d1), Vec4(1, 1, 0, 1), Vec4(0, 1, 0, 1)]
only right triangles are rendered using this glsl shader (on left is fixed pipeline):
void main()
{
gl_FragColor = texture2D(colormap, vec2(gl_TexCoord[0].x / glTexCoord[0].w, gl_TexCoord[0].y);
}
so.. only U is perspective and V is linear.
I am trying to make a semicircle with a straight line going from its end point to point at a target. I have tried multiple ways for a day now and can't get it to point exactly at a target position. Here is my progress so far:
I'm trying to get the dark green line to go trough the red point on the yellow line.
Posting the code so far:
vector init = <105.45535, 105.83867, 2239.99976>;
vector init_unit = <-0.54465, 0.83867, 0.00000>;
vector target = <106,104,2241>;
default{
state_entry(){
llListen(-215485231, "", NULL_KEY, "");
}
listen(integer c, string n, key i, string m){
list temp = llParseString2List(m, ["|"], []);
init = (vector)llList2String(temp, 0); //position
init_unit = (vector)llList2String(temp, 1);
init_unit = llVecNorm(<init_unit.x, init_unit.y, 0.0>); //line norm vector
float angle = llAtan2(init_unit.y, init_unit.x); //find angle
rotation delta = llEuler2Rot(<0.0, -PI_BY_TWO, PI>); //extra rotation
rotation rot = delta * llEuler2Rot(<0.0, 0.0, angle>); //convert from vector to rotation
init = init + <0.0, -0.45, 0.0>*rot; //make new offset
vector p1 = target - init;
float angle2 = llAtan2(p1.y, p1.x);
rotation rot2 = llEuler2Rot(<0.0, 0.0, angle>);
vector p2 = init + (<0.0, 0.45, 0.0>*(delta*rot2)); //find last other side of semi circle
p2 = (p2 + p1) - init;
float angle3 = llAcos((p1*p2)/(llVecMag(p1)*llVecMag(p2))); //find angle between vectors
llSetRot(delta * llEuler2Rot(<0.0, 0.0, angle2+angle3>)); //set rotation
llSetPos(init); //set postion
}
}
The semicircle is 1m on y axis and middle on each end is +/- <0,0.45,0>.
Please ask if anything is unclear.