I was wondering if anyone can help me. I'm trying to modal an oval room, and the joists run parallel to one another at 400mm intervals, starting and finishing 200mm from the apexes of the oval. The central joist falls on the centre of the oval at (0,0).
So the oval is positioned at angle = 0, with a centre of (0,0). The major axis is 6000mm long in the x-direction and the minor axis is 3500mm long in the y-direction. The joists run in the y-direction too.
I need to find out the node for each joist along the outside edge of the ellipse. So obviously, I know the x values will be -2800, -2400, ..., 0, ..., 2000, 2400, 2800, and that the central joist will have one node at (0, 1750) and one at (0, -1750), but how can I find the y values for all the other x co-ordinates?
Many thanks.
p.s. In case you can't tell I have exceedinly rudimentary MATLAB skills.
It's convenient to work with semi-axes, denoted a and b below. The equation of ellipse is (x/a)^2+(y/b)^2=1, which gives two values of y, positive b*sqrt(1-(x./a)^2) and negative b*sqrt(1-(x./a)^2).
In MATLAB you can compute them this way:
a = 6000/2;
b = 3500/2;
x = -2800:400:2800;
yP = b.*sqrt(1-(x./a).^2);
yN = - yP;
So, yP contains the positive y-coordinates and yN contains negative y-coordinates.
The dots in front of arithmetic operations mean they are performed on vectors componentwise.
Related
Having this coordinate system:
And this dominant vertical vanishing point:
I would like to rotate the image around x axis so the vanishing point is at infinity. That means that all vertical lines are parallel.
I am using matlab. I find the line segmentes using LSD and the vanishing point using homogeneous coordinates. I would like to use angle-axis representation, then convert it to a rotation matrix and pass this to imwarp and get the rotated image. Also would be good to know how to rotate the segments. The segments are as (x1,y1,x2,y2).
Image above example:
Vanishin point in homogenous coordinates:
(x,y,z) = 1.0e+05 * [0.4992 -2.2012 0.0026]
Vanishin point in cartesian coordinates (what you see in the image):
(x,y) = [190.1335 -838.3577]
Question: With this vanishing point how do I compute the rotation matrix in the world x axis as explained above?
If all you're doing is rotating the image so that the vector from the origin to the vanishing point, is instead pointing directly vertical, here's an example.
I = imread('cameraman.tif');
figure;imagesc(I);set(gcf,'colormap',gray);
vp=-[190.1335 -838.3577,0]; %3d version,just for cross-product use,-ve ?
y=[0,1,0]; %The vertical axis on the plot
u = cross(vp,y); %you know it's going to be the z-axis
theta = -acos(dot(vp/norm(vp),y)); %-ve ?
rotMat = vrrotvec2mat([u, theta]);
J=imwarp(I,affine2d (rotMat));
figure;imagesc(J);set(gcf,'colormap',gray); %tilted image
You can play with the negatives, and plotting, since I'm not sure about those parts applying to your situation. The negatives may come from plotting upside down, or from rotation of the world vs. camera coordinate system, but I don't have time to think about it right now.
EDIT
If you want to rotation about the X-axis, this might work (adapted from https://www.mathworks.com/matlabcentral/answers/113074-how-to-rotate-an-image-along-y-axis), or check out: Rotate image over X, Y and Z axis in Matlab
[rows, columns, numberOfColorChannels] = size(I);
newRows = rows * cos(theta);
rotatedImage = imresize(I, [newRows, columns]);
We have these logarithmic spirals which are circling around the centre of the coordinate system:
x = ebθ cos(θ)
y = ebθ sin(θ)
where the ebθ is the distance between the point (which is on the spiral) and the centre; and the θ is the angle between the line connecting the point and the origin and the axis x.
Consider a spiral where the angle is θ ϵ <0,10π> and the parameter is b=0.1. By thickening points on the spirals (and the angle θ) calculate the circumference with the relative precision better than 1%. Draw the spiral!
I'm preparing for a (MATLAB) test and I'm stuck with this exercise. Please help, any hint is appreciated.
Start by computing a list of x,y for your range of theta and value of b. For more accurate results, have your theta increment in smaller steps (I chose 5000 arbitrarily). Then, its simply computing the distance for each pair of consecutive points and summing them up.
t = linspace(0,10*pi,5000);
b = 0.1;
x = exp(b*t).*cos(t);
y = exp(b*t).*sin(t);
result = sum(sqrt((x(2:end) - x(1:end-1)).^2 + (y(2:end)-y(1:end-1)).^2))
I have a surface Z on a X-Y grid for which I want to find the intersection point with a line. I used so far this code for finding the intersection:
x_ray = x_source + t * x_dir
y_ray = y_source + t * y_dir
z_ray = z_source + t * z_dir
height_above_plane = #(t) z_source + t * z_dir - interp2(X, Y, Z, ...
x_source + t*x_dir, y_source + t*y_dir)
t_intercept = fzero(height_above_plane, 0);
my problem is that when my surface is "wiggly", the function has several zero crossing points, and I want to find the minimal out of them.
How can I do that?
Thanks
A possible approach is to project the ray onto the XY domain and draw the corresponding Bresenham line. As you go along this line, grid cell per grid cell, you will compute the Z altitudes along the ray and check if their range overlaps the range of altitudes of the surface (i.e. the min and max value in this cell).
If yes, you have to find the 3D intersection between the ray and the interpolating surface, an hyperbolic paraboloid. If the intersection does fall inside the grid cell considered, you are done. Otherwise, continue the march along the ray.
Convert the surface to matlab mesh, then use this code.
I wasn't sure how to word this question so please link me to any answer if this has been asked before.
Let's say I have a graph with points making a line that starts at (5, 10) and goes to (10,10), but I want to move the points so that the first point starts at (0, 10) up to (5, 10). How do I go about doing this? Or what is this called so I can search on my own? I still want the points to be the same distances apart relative to each other but with one of the points at a specific location that I specify.
Simply take all of your points, and subtract or add them by a certain amount to move the origin. As such, because you want to move your line such that the horizontal component is shifted to the left by 5, you can just subtract every x co-ordinate by 5.
As such, assuming your co-ordinates are in x and y, just do:
final_x = x - 5;
final_y = y;
Then go ahead and plot these values:
plot(final_x, final_y);
In general, if you want to move your points by a prescribed amount, do this:
final_x = x + x_shift;
final_y = y + y_shift;
x_shift and y_shift would be the amount of movement you want the x and y co-ordinates to move. In this case, you want to move everything to the left by 5, and so x_shift = -5 and y_shift = 0. If you want to move everything such that the origin is located at (0,0), you would make x_shift and y_shift to be the minimum of the x and y values, or:
x_shift = min(x);
y_shift = min(y);
Using this will ensure that all of your points are with respect to (0,0).
I need to make a small program that draws three circles, a line between the first two, and then determines if the third touches or intersects the line. I have done everything but the last part. I am trying to use the points to determine if the area is 0, which would mean that the third point is, in fact, intersecting the line. Right? Or I could use another way. Technically the third circle can be within 3 pixels of the line. The problem is near the bottom at the hashtag. I would appreciate any help or suggestions that move this in another direction. Thank you.
import turtle
x1, y1 = eval(input("Enter coordinates for the first point x, y: "))
x2, y2 = eval(input("Enter coordinates for the second point x, y: "))
x3, y3 = eval(input("Enter coordinates for the third point x, y: "))
turtle.penup()
turtle.goto(x1, y1)
turtle.pendown()
turtle.circle(3)
turtle.penup()
turtle.goto(x2, y2)
turtle.pendown()
turtle.circle(3)
turtle.penup()
turtle.goto(x3, y3)
turtle.pendown()
turtle.circle(3)
turtle.penup()
turtle.color("red")
turtle.goto(x1, y1)
turtle.pendown()
turtle.goto(x2, y2)
a = (x1, y1)
c = (x3, y3)
#can't multiply sequence by non-int of type 'tuple'
area = (a * c) / 2
if area == 0:
print("Hit")
else:
print("Miss")
Ther center of 3rd circle is (x3,y3) and have a radius 3 and you are trying to determine any intersection with ([x1,y1],[x2,y2]) line segment.
If any point in the line is within the circle, then there is an intersection.
Circle region formula is: (x-x3)^2 + (y-y3)^2 < 3^2
You should test for every point on the line whether this inequality holds and if any single point satisfies this condition, then you can conclude that the line and circle intersect.
The first step would be to determine coordinate points of line segment (all points between [x1,y1],[x2,y2] points in a straight line) then you can try to test these points in a loop.
You can calculate the area of the triangle by defining vectors from one vertex to the other two (adding a third constant coordinate to embed the plane in 3-dimensional space (so cross-products make sense)),
#pseudocode
b = (x2, y2, 1) - (x1, y1, 1) = (x2-x1, y2-y1, 0)
c = (x3, y3, 1) - (x1, y1, 1) = (x3-x1, y3-y1, 0)
then take the cross-product of these,
a = b cross c = (by*cz-bz*cy, bz*cx-bx*cz, bx*cy-by*cx)
then take the magnitude of this resulting vector which is the area of the parallelogram defined by the two vectors,
pa = |a| = ax^2 + ay^2 + az^2
then divide by two to get the area of the triangle (half of the parallelogram).
ta = pa/2
Source: http://en.wikipedia.org/wiki/Triangle_area#Using_vectors
Am I wright? The position of the circles to each other does not matter?
Make a linear function from the line between the two center points. (ax+b=y)
Where a is the gradient and b is the y-intersection.
To rotate a through 90° is easy. Inverse and negate a.
Find b of the second linear function. b'=y-a'*x .
At once replace the x,y with the coordinates of your 3. circle point.
Now you have a linear function which is rectangular to the old one and where the third circle point is part of.
Intersect the old linear function with the new one.
You'll get the lot point.
You need to find out the distance between the 3. circle point and the lot point and whether it is greater than the radius.
You need there functions (JS):
function makelinear (x1,y1,x2,y2){
var temp=x2-x1;
if(temp==0)temp=0.00000000000001;//not clean but fast.
var a=(y2-y1)/temp,
b=y1-a*x1;
return[a,b];
}
function ninetydeg(a,b,x,y){
var aout=1/a,
bout=y+aout*x;
return [aout,bout];
}
function lineintersection(a1,b1,a2,b2){
var temp=a1-a2;
if(temp==0)temp=0.00000000000001;
var x=(b2-b1)/temp,
y=a1*x+b1;
return[x,y];
}
function distance(x1,y1,x2,y2){
var x=x1-x2,
y=y1-y2;
return(Math.sqrt(x*x+y*y));
}
Sorry for to be complicate, I've found no other solution in short time. May be there is a vector solution.