I'm trying to draw an ellipse between two points. So far, I have it mostly working:
The issue comes with setting the ellipse height (ellipse_h below).
x = center_x + radius*np.cos(theta+deg)
y = center_y - ellipse_h * radius*np.sin(theta+deg)
In this example, it's set to -0.5:
Can anyone please help me rotate the ellipse height with the ellipse? Thank you!
import numpy as np
import matplotlib.pyplot as plt
def distance(x1, y1, x2, y2):
return np.sqrt(np.power(x2 - x1, 2) + np.power(y2 - y1, 2) * 1.0)
def midpoint(x1, y1, x2, y2):
return [(x1 + x2) / 2,(y1 + y2) / 2]
def angle(x1, y1, x2, y2):
#radians
return np.arctan2(y2 - y1, x2 - x1)
x1 = 100
y1 = 150
x2 = 200
y2 = 190
ellipse_h = -1
x_coords = []
y_coords = []
mid = midpoint(x1, y1, x2, y2)
center_x = mid[0]
center_y = mid[1]
ellipse_resolution = 40
step = 2*np.pi/ellipse_resolution
radius = distance(x1, y1, x2, y2) * 0.5
deg = angle(x1, y1, x2, y2)
cos = np.cos(deg * np.pi /180)
sin = np.sin(deg * np.pi /180)
for theta in np.arange(0, np.pi+step, step):
x = center_x + radius*np.cos(theta+deg)
y = center_y - ellipse_h * radius*np.sin(theta+deg)
x_coords.append(x)
y_coords.append(y)
plt.xlabel("X")
plt.ylabel("Y")
plt.title("Arc between 2 Points")
plt.plot(x_coords,y_coords)
plt.scatter([x1,x2],[y1,y2])
plt.axis('equal')
plt.show()
A simple solution is to describe the ellipse by its standard parametric equation, as you effectively did. However, under the assumption that it is centered on the origin of the coordinate system, it becomes straightforward to then apply a rotation to its points using a 2d rotation matrix and finally apply a translation to position it on its true center. This gives the following:
import numpy as np
import matplotlib.pyplot as plt
# extreme points along the major axis
x1, y1 = 100, 150
x2, y2 = 200, 190
# along minor axis
height = 15
# number of points
n = 100
# center
x_c, y_c = (x1 + x2)/2, (y1 + y2)/2
# width (major axis) and height (minor) of the ellipse halved
a, b = np.sqrt((x2 - x1)**2 + (y2 - y1)**2)/2, height/2
# rotation angle
angle = np.arctan2(y2 - y1, x2 - x1)
# standard parametric equation of an ellipse
t = np.linspace(0, 2*np.pi, n)
ellipse = np.array([a*np.cos(t), b*np.sin(t)])
# 2d rotation matrix
R = np.array([[np.cos(angle), -np.sin(angle)],
[np.sin(angle), np.cos(angle)]])
# apply the rotation to the ellipse
ellipse_rot = R # ellipse
plt.plot(x_c + ellipse_rot[0], y_c + ellipse_rot[1], 'r' )
plt.scatter([x1, x2], [y1, y2], color='k')
plt.axis('equal')
plt.show()
See the output for different heights:
Following your comment, for the limiting case of the circle, you need to specify height = np.sqrt((x2 - x1)**2 + (y2 - y1)**2), so that a = b.
Hope this helps !
Trying to get my my head around this program we need to create
What is needed is as per the notes:
create a function named
arbitraryMirror() that allows the user to place a mirror at an arbitrary angle, causing an intersect and therefore mirror the image.
This will need to be done on either a square or rectangle picture.
As per the pics below, this is the Output of what is required.
Output
I know how to mirror a pic (as shown below) with a square image, but i cannot work out if this can also be done with a rectangle image?
Cross
I had a look at a method of using y=mx+b but it seems overcomplicated?
Maybe there is some coordinate geometry i need? Or algebra?
Any help would be greatly appreciated!
The key formulas are (python):
# (x0, y0) and (x1, y1) are two points on the mirroring line
# dx, dy, L is the vector and lenght
dx, dy = x1 - x0, y1 - y0
L = (dx**2 + dy**2) ** 0.5
# Tangent (tx, ty) and normal (nx, ny) basis unit vectors
tx, ty = dx / L, dy / L
nx, ny = -dy / L, dx / L
# For each pixel
for y in range(h):
for x in range(w):
# Map to tangent/normal space
n = (x+0.5 - x0)*nx + (y+0.5 - y0)*ny
t = (x+0.5 - x0)*tx + (y+0.5 - y0)*ty
# If we're in the positive half-space
if n >= 0:
# Compute mirrored point in XY space
# (negate the normal component)
xx = int(x0 + t*tx - n*nx + 0.5)
yy = int(y0 + t*ty - n*ny + 0.5)
# If valid copy to destination
if 0 <= xx < w and 0 <= yy < h:
img[y][x] = img[yy][xx]
Here you can see an example of the results
The top-left red corner are pixels that would be mirroring pixels outside of the original image and they're left untouched by the above code.
I used to draw lines (given some start and end points) at pygame like this: pygame.draw.line(window, color_L1, X0, X1, 2), where 2 was defining the thickness of the line.
As, anti-aliasing is not supported by .draw, so I moved to .gfxdraw and pygame.gfxdraw.line(window, X0[0], X0[1], X1[0], X1[1], color_L1).
However, this does not allow me to define the thickness of the line. How could I have thickness and anti-aliasing together?
After many trials and errors, the optimal way to do it would be the following:
First, we define the center point of the shape given the X0_{x,y} start and X1_{x,y} end points of the line:
center_L1 = (X0+X1) / 2.
Then find the slope (angle) of the line:
length = 10 # Total length of line
thickness = 2
angle = math.atan2(X0[1] - X1[1], X0[0] - X1[0])
Using the slope and the shape parameters you can calculate the following coordinates of the box ends:
UL = (center_L1[0] + (length/2.) * cos(angle) - (thickness/2.) * sin(angle),
center_L1[1] + (thickness/2.) * cos(angle) + (length/2.) * sin(angle))
UR = (center_L1[0] - (length/2.) * cos(angle) - (thickness/2.) * sin(angle),
center_L1[1] + (thickness/2.) * cos(angle) - (length/2.) * sin(angle))
BL = (center_L1[0] + (length/2.) * cos(angle) + (thickness/2.) * sin(angle),
center_L1[1] - (thickness/2.) * cos(angle) + (length/2.) * sin(angle))
BR = (center_L1[0] - (length/2.) * cos(angle) + (thickness/2.) * sin(angle),
center_L1[1] - (thickness/2.) * cos(angle) - (length/2.) * sin(angle))
Using the computed coordinates, we draw an unfilled anti-aliased polygon (thanks to #martineau) and then fill it as suggested in the documentation of pygame's gfxdraw module for drawing shapes.
pygame.gfxdraw.aapolygon(window, (UL, UR, BR, BL), color_L1)
pygame.gfxdraw.filled_polygon(window, (UL, UR, BR, BL), color_L1)
I would suggest a filled rectangle, as shown here: https://www.pygame.org/docs/ref/gfxdraw.html#pygame.gfxdraw.rectangle.
Your code would look something like:
thickLine = pygame.gfxdraw.rectangle(surface, rect, color)
and then remember to fill the surface. This is along the lines of:
thickLine.fill()
You can also do a bit of a hack with the pygame.draw.aalines() function by drawing copies of the line +/- 1-N pixels around the original line (yes, this isn't super efficient, but it works in a pinch). For example, assuming we have a list of line segments (self._segments) to draw and with a width (self._LINE_WIDTH):
for segment in self._segments:
if len(segment) > 2:
for i in xrange(self._LINE_WIDTH):
pygame.draw.aalines(self._display, self._LINE_COLOR, False,
((x,y+i) for x,y in segment))
pygame.draw.aalines(self._display, self._LINE_COLOR, False,
((x,y-i) for x,y in segment))
pygame.draw.aalines(self._display, self._LINE_COLOR, False,
((x+i,y) for x,y in segment))
pygame.draw.aalines(self._display, self._LINE_COLOR, False,
((x-i,y) for x,y in segment))
Your answer gets the job done but I think this would be a better/more readable way to do it. This is piggybacking off of your answer though so credit to you.
from math import atan2, cos, degrees, radians, sin
def Move(rotation, steps, position):
"""Return coordinate position of an amount of steps in a direction."""
xPosition = cos(radians(rotation)) * steps + position[0]
yPosition = sin(radians(rotation)) * steps + position[1]
return (xPosition, yPosition)
def DrawThickLine(surface, point1, point2, thickness, color):
angle = degrees(atan2(point1[1] - point2[1], point1[0] - point2[0]))
vertices = list()
vertices.append(Move(angle-90, thickness, point1))
vertices.append(Move(angle+90, thickness, point1))
vertices.append(Move(angle+90, thickness, point2))
vertices.append(Move(angle-90, thickness, point2))
pygame.gfxdraw.aapolygon(surface, vertices, color)
pygame.gfxdraw.filled_polygon(surface, vertices, color)
Keep in mind that this treats the thickness more as a radius than a diameter. If you want it to act more like a diameter you can divide each instance of the variable by 2.
So anyway, this calculates all the points of the rectangle and fills it in. It does this by going to each point and calculating the two adjacent points by turning 90 degrees and moving forward.
Here is a slightly faster and shorter solution:
def drawLineWidth(surface, color, p1, p2, width):
# delta vector
d = (p2[0] - p1[0], p2[1] - p1[1])
# distance between the points
dis = math.hypot(*d)
# normalized vector
n = (d[0]/dis, d[1]/dis)
# perpendicular vector
p = (-n[1], n[0])
# scaled perpendicular vector (vector from p1 & p2 to the polygon's points)
sp = (p[0]*width/2, p[1]*width/2)
# points
p1_1 = (p1[0] - sp[0], p1[1] - sp[1])
p1_2 = (p1[0] + sp[0], p1[1] + sp[1])
p2_1 = (p2[0] - sp[0], p2[1] - sp[1])
p2_2 = (p2[0] + sp[0], p2[1] + sp[1])
# draw the polygon
pygame.gfxdraw.aapolygon(surface, (p1_1, p1_2, p2_2, p2_1), color)
pygame.gfxdraw.filled_polygon(surface, (p1_1, p1_2, p2_2, p2_1), color)
The polygon's points here are calculated using vector math rather than trigonometry, which is much less costly.
If efficiency is of the essence, it's easy to further optimize this code - for instance the first few lines can be condensed to:
d = (p2[0] - p1[0], p2[1] - p1[1])
dis = math.hypot(*d)
sp = (-d[1]*width/(2*dis), d[0]*width/(2*dis))
Hope this helps someone.
This is a slightly longer code, but maybe will help someone.
It uses vectors and create a stroke on each side of the line connecting two points.
def make_vector(pointA,pointB): #vector between two points
x1,y1,x2,y2 = pointA[0],pointA[1],pointB[0],pointB[1]
x,y = x2-x1,y2-y1
return x,y
def normalize_vector(vector): #sel explanatory
x, y = vector[0], vector[1]
u = math.sqrt(x ** 2 + y ** 2)
try:
return x / u, y / u
except:
return 0,0
def perp_vectorCL(vector): #creates a vector perpendicular to the first clockwise
x, y = vector[0], vector[1]
return y, -x
def perp_vectorCC(vector): #creates a vector perpendicular to the first counterclockwise
x, y = vector[0], vector[1]
return -y, x
def add_thickness(point,vector,thickness): #offsets a point by the vector
return point[0] + vector[0] * thickness, point[1] + vector[1] * thickness
def draw_line(surface,fill,thickness, start,end): #all draw instructions
x,y = make_vector(start,end)
x,y = normalize_vector((x,y))
sx1,sy1 = add_thickness(start,perp_vectorCC((x,y)),thickness//2)
ex1,ey1 = add_thickness(end,perp_vectorCC((x,y)),thickness//2)
pygame.gfxdraw.aapolygon(surface,(start,end,(ex1,ey1),(sx1,sy1)),fill)
pygame.gfxdraw.filled_polygon(surface, (start, end, (ex1, ey1), (sx1, sy1)), fill)
sx2, sy2 = add_thickness(start, perp_vectorCL((x, y)), thickness // 2)
ex2, ey2 = add_thickness(end, perp_vectorCL((x, y)), thickness//2)
pygame.gfxdraw.aapolygon(surface, (start, end, (ex2, ey2), (sx2, sy2)), fill)
pygame.gfxdraw.filled_polygon(surface, (start, end, (ex2, ey2), (sx2, sy2)), fill)
I've been trying to rotate a bunch of lines by 90 degrees (that together form a polyline). Each line contains two vertices, say (x1, y1) and (x2, y2). What I'm currently trying to do is rotate around the center point of the line, given center points |x1 - x2| and |y1 - y2|. For some reason (I'm not very mathematically savvy) I can't get the lines to rotate correctly.
Could someone verify that the math here is correct? I'm thinking that it could be correct, however, when I set the line's vertices to the new rotated vertices, the next line may not be grabbing the new (x2, y2) vertex from the previous line, causing the lines to rotate incorrectly.
Here's what I've written:
def rotate_lines(self, deg=-90):
# Convert from degrees to radians
theta = math.radians(deg)
for pl in self.polylines:
self.curr_pl = pl
for line in pl.lines:
# Get the vertices of the line
# (px, py) = first vertex
# (ox, oy) = second vertex
px, ox = line.get_xdata()
py, oy = line.get_ydata()
# Get the center of the line
cx = math.fabs(px-ox)
cy = math.fabs(py-oy)
# Rotate line around center point
p1x = cx - ((px-cx) * math.cos(theta)) - ((py-cy) * math.sin(theta))
p1y = cy - ((px-cx) * math.sin(theta)) + ((py-cy) * math.cos(theta))
p2x = cx - ((ox-cx) * math.cos(theta)) - ((oy-cy) * math.sin(theta))
p2y = cy - ((ox-cx) * math.sin(theta)) + ((oy-cy) * math.cos(theta))
self.curr_pl.set_line(line, [p1x, p2x], [p1y, p2y])
The coordinates of the center point (cx,cy) of a line segment between points (x1,y1) and (x2,y2) are:
cx = (x1 + x2) / 2
cy = (y1 + y2) / 2
In other words it's just the average, or arithmetic mean, of the two pairs of x and y coordinate values.
For a multi-segmented line, or polyline, its logical center point's x and y coordinates are just the corresponding average of x and y values of all the points. An average is just the sum of the values divided by the number of them.
The general formulas to rotate a 2D point (x,y) θ radians around the origin (0,0) are:
x′ = x * cos(θ) - y * sin(θ)
y′ = x * sin(θ) + y * cos(θ)
To perform a rotation about a different center (cx, cy), the x and y values of the point need to be adjusted by first subtracting the coordinate of the desired center of rotation from the point's coordinate, which has the effect of moving (known in geometry as translating) it is expressed mathematically like this:
tx = x - cx
ty = y - cy
then rotating this intermediate point by the angle desired, and finally adding the x and y values of the point of rotation back to the x and y of each coordinate. In geometric terms, it's the following sequence of operations: Tʀᴀɴsʟᴀᴛᴇ ─► Rᴏᴛᴀᴛᴇ ─► Uɴᴛʀᴀɴsʟᴀᴛᴇ.
This concept can be extended to allow rotating a whole polyline about any arbitrary point—such as its own logical center—by just applying the math described to each point of each line segment within it.
To simplify implementation of this computation, the numerical result of all three sets of calculations can be combined and expressed with a pair of mathematical formulas which perform them all simultaneously. So a new point (x′,y′) can be obtained by rotating an existing point (x,y), θ radians around the point (cx, cy) by using:
x′ = ( (x - cx) * cos(θ) + (y - cy) * sin(θ) ) + cx
y′ = ( -(x - cx) * sin(θ) + (y - cy) * cos(θ) ) + cy
Incorporating this mathematical/geometrical concept into your function produces the following:
from math import sin, cos, radians
def rotate_lines(self, deg=-90):
""" Rotate self.polylines the given angle about their centers. """
theta = radians(deg) # Convert angle from degrees to radians
cosang, sinang = cos(theta), sin(theta)
for pl in self.polylines:
# Find logical center (avg x and avg y) of entire polyline
n = len(pl.lines)*2 # Total number of points in polyline
cx = sum(sum(line.get_xdata()) for line in pl.lines) / n
cy = sum(sum(line.get_ydata()) for line in pl.lines) / n
for line in pl.lines:
# Retrieve vertices of the line
x1, x2 = line.get_xdata()
y1, y2 = line.get_ydata()
# Rotate each around whole polyline's center point
tx1, ty1 = x1-cx, y1-cy
p1x = ( tx1*cosang + ty1*sinang) + cx
p1y = (-tx1*sinang + ty1*cosang) + cy
tx2, ty2 = x2-cx, y2-cy
p2x = ( tx2*cosang + ty2*sinang) + cx
p2y = (-tx2*sinang + ty2*cosang) + cy
# Replace vertices with updated values
pl.set_line(line, [p1x, p2x], [p1y, p2y])
Your center point is going to be:
centerX = (x2 - x1) / 2 + x1
centerY = (y2 - y1) / 2 + y1
because you take half the length (x2 - x1) / 2 and add it to where your line starts to get to the middle.
As an exercise, take two lines:
line1 = (0, 0) -> (5, 5)
then: |x1 - x2| = 5, when the center x value is at 2.5.
line2 = (2, 2) -> (7, 7)
then: |x1 - x2| = 5, which can't be right because that's the center for
the line that's parallel to it but shifted downwards and to the left
If I know the center(x,y,z) of the arc and the diameter, and the starting and ending point, how can I generate the values between the start and the end?
It sounds like your "arc" is an circular approximation to a curve between two known points. I guessing this from the word "diameter" (which is twice the radius) in your post. To do this you parameterize the circle (t) -> (x,y) where t goes from 0..2pi. Given a center, two end points and a radius we can approximate a portion of the curve like this:
from numpy import cos,sin,arccos
import numpy as np
def parametric_circle(t,xc,yc,R):
x = xc + R*cos(t)
y = yc + R*sin(t)
return x,y
def inv_parametric_circle(x,xc,R):
t = arccos((x-xc)/R)
return t
N = 30
R = 3
xc = 1.0
yc = 3.0
start_point = (xc + R*cos(.3), yc + R*sin(.3))
end_point = (xc + R*cos(2.2), yc + R*sin(2.2))
start_t = inv_parametric_circle(start_point[0], xc, R)
end_t = inv_parametric_circle(end_point[0], xc, R)
arc_T = np.linspace(start_t, end_t, N)
from pylab import *
X,Y = parametric_circle(arc_T, xc, yc, R)
plot(X,Y)
scatter(X,Y)
scatter([xc],[yc],color='r',s=100)
axis('equal')
show()
This example is only in 2D, but it is easily adaptable since the curve will always lie along the plane between the two points and the center.