How do I find the maximum rounding I can apply to either corner for any amount of rounding on the other corner?
Answers to questions from the comments:
1) The inner and outer large arcs (those that are 90 degrees wide here) always have the same center
2) When asking for the maximum rounding that you can do, what are the constraints on the other, smaller circle? Does it need to be at least some radius? Otherwise you are doing to end up with just one rounding.
One of the two rounding circle's radius is given. There are no other constraints other than the maximum of the other circle which I just can't find.
If the "fixed" corner that I refer to has zero rounding then I'm searching for the maximum rouding that can be applied with only the other corner.
3) What constitutes as the maximum rounding? Are you trying to choose between the two examples above? Or is finding either of those cases considered a solution?
Either of the shown cases is a perfect solution. E.g. in the first image the the radius of the smaller circle might be given. Then I'm looking for the maximum radius of the larger one.
These images are just examples for perfect solutions.
4) is there any constraints on the two arcs? What happens if the arcs can't fit a full circle? Would the answer be the largest that fits?
How exactly do you mean that the arcs can't fit a full circle?
The all circles are perfect circles, but I can't figure out the max size of the rounding possible, or how to calculate it's position. Here's some images that describe the problem.
Given that the origin of the coordinate system is at the center point of the inner and outer large arcs...
For the first case where the large circle is tangent to the outer edge, the center point of the large circle is
x = R cos(t) / (1 + cos(t))
y = R sin(t) / (1 + cos(t))
where R is the radius of the outer arc segment, and t is the angle between the x-axis and the ray from the origin through the center of the large circle.
For the second case where the large circle is tangent to the inner edge, the center point of the large circle is
x = R cos(t) / (1 - cos(t))
y = R sin(t) / (1 - cos(t))
where R is the radius of the inner arc segment, and t is the angle...
In both cases, the radius of the circle is equal to its x coordinate. The range of t is between some minimum angle and PI/2. At PI/2, the circle is vanishingly small. At the minimum angle, the y value is equal to the opposite radius. In other words, for the first case where the large circle is tangent to the outer edge, the minimum angle is such that y is equal to the inner radius. Whereas if the circle is tangent to the inner edge, the minimum angle is such that y is equal to the outer radius. It can be proven mathematically that the minimum angle is the same for both cases (tangent to inner and tangent to outer both have the same minimum angle for a given inner and outer radius). However, computing the minimum angle is a bit of a challenge. The only way I know how to do it is by playing the high/low game, e.g.
- (CGFloat)computeAngleForOuterTangentGivenY:(CGFloat)Y
{
CGFloat y;
double high = M_PI_2;
double low = 0;
double mid = M_PI_4;
while ( high - low > 1e-9 )
{
y = (self.outerRadius * sin( mid )) / (1.0 + cos( mid ));
if ( y > Y )
high = mid;
else
low = mid;
mid = (low + high) / 2.0;
}
return( mid );
}
- (CGFloat)computeAngleForInnerTangentGivenY:(CGFloat)Y
{
CGFloat y;
double high = M_PI_2;
double low = 0;
double mid = M_PI_4;
while ( high - low > 1e-9 )
{
y = (self.innerRadius * sin( mid )) / (1.0 - cos( mid ));
if ( y > Y )
low = mid;
else
high = mid;
mid = (low + high) / 2.0;
}
return( mid );
}
It takes about 30 passes for the loop to converge to an answer.
To find the coordinates of the small circle, note that the small circle has the same y value as the large circle, and is tangent to the opposite edge of the arc segment. Therefore, compute the angle t for the small circle based on its y value using the appropriate high/low algorithm, and then compute the x value using the formulas above.
QED
The question isn't posed correctly without showing both ends of the line segment. Suppose for a moment that each line segment is a data structure that maintains not only the end points, but also cap radius in each point, and also knows the angle going out to the next endpoint that this line will attach to. Each cap radius will subtract from the length of the line segment that has to be stroked as a rectangle. Assume you have a line of interest between points B and C, where B joins to another (longer) segment A, and C joins to another (longer) segment D. If line BC is length 10, with cap radius B and cap radius C both set to 4, then you will only render rectangle of length 2 for the straight part of the line segment, while length 4 is used to draw the arc to A, and another length 4 is used to draw the arc to D.
Furthermore, the maximum cap radius for C is constrained not only by BC and B's cap radius, but also by CD and D's cap radius.
Related
What is the algorithm used in OpenCV function convexityDefects() to calculate the convexity defects of a contour?
Please, describe and illustrate the high-level operation of the algorithm, along with its inputs and outputs.
Based on the documentation, the input are two lists of coordinates:
contour defining the original contour (red on the image below)
convexhull defining the convex hull corresponding to that contour (blue on the image below)
The algorithm works in the following manner:
If the contour or the hull contain 3 or less points, then the contour is always convex, and no more processing is needed. The algorithm assures that both the contour and the hull are accessed in the same orientation.
N.B.: In further explanation I assume they are in the same orientation, and ignore the details regarding representation of the floating point depth as an integer.
Then for each pair of adjacent hull points (H[i], H[i+1]), defining one edge of the convex hull, calculate the distance from the edge for each point on the contour C[n] that lies between H[i] and H[i+1] (excluding C[n] == H[i+1]). If the distance is greater than zero, then a defect is present. When a defect is present, record i, i+1, the maximum distance and the index (n) of the contour point where the maximum located.
Distance is calculated in the following manner:
dx0 = H[i+1].x - H[i].x
dy0 = H[i+1].y - H[i].y
if (dx0 is 0) and (dy0 is 0) then
scale = 0
else
scale = 1 / sqrt(dx0 * dx0 + dy0 * dy0)
dx = C[n].x - H[i].x
dy = C[n].y - H[i].y
distance = abs(-dy0 * dx + dx0 * dy) * scale
It may be easier to visualize in terms of vectors:
C: defect vector from H[i] to C[n]
H: hull edge vector from H[i] to H[i+1]
H_rot: hull edge vector H rotated 90 degrees
U_rot: unit vector in direction of H_rot
H components are [dx0, dy0], so rotating 90 degrees gives [-dy0, dx0].
scale is used to find U_rot from H_rot, but because divisions are more computationally expensive than multiplications, the inverse is used as an optimization. It's also pre-calculated before the loop over C[n] to avoid recomputing each iteration.
|H| = sqrt(dx0 * dx0 + dy0 * dy0)
U_rot = H_rot / |H| = H_rot * scale
Then, a dot product between C and U_rot gives the perpendicular distance from the defect point to the hull edge, and abs() is used to get a positive magnitude in any orientation.
distance = abs(U_rot.C) = abs(-dy0 * dx + dx0 * dy) * scale
In the scenario depicted on the above image, in first iteration, the edge is defined by H[0] and H[1]. The contour points tho examine for this edge are C[0], C[1], and C[2] (since C[3] == H[1]).
There are defects at C[1] and C[2]. The defect at C[1] is the deepest, so the algorithm will record (0, 1, 1, 50).
The next edge is defined by H[1] and H[2], and corresponding contour point C[3]. No defect is present, so nothing is recorded.
The next edge is defined by H[2] and H[3], and corresponding contour point C[4]. No defect is present, so nothing is recorded.
Since C[5] == H[3], the last contour point can be ignored -- there can't be a defect there.
I am doing some computer vision based hand gesture recognising stuff. Here, I want to detect a circle (a circular motion) made by my hand. My initial stages are working fine and I am able to get a blob whose centroid from each frame I am plotting. This is essentially my data set. A collection of 2D co-ordinate points. Now I want to detect a circular type motion and say generate a call to a function which says "Circle Detected". The circle detector will give a YES / NO boolean output.
Here is a sample of the data set I am generating in 40 frames
The x, y values are just plotted to a bitmap image using MATLAB.
My initial hand movement was slow and later I picked up speed to complete the circle within stipulated time (40 frames). There is no hard and fast rule about the number of frames thing but for now I am using a 40 frame sliding window for circle detection (0-39) then (1-40) then (2-41) etc.
I am also calculating the arc-tangent between successive points using:
angle = atan2(prev_y - y, prev_x - x) * 180 / pi;
Now what approach should I take for detecting a circle (This sample image should result in a YES). The angle as I am noticing is not steadily increasing from 0 to 360. It does increase but with jumps here and there.
If you are only interested in full or nearly full circles:
I think that the standard parameter estimation approach: Hough/RANSAC won't work very well in this case.
Since you have frames order and therefore distances between consecutive blob centers, you can create a nearly uniform sub sample of the data (let say, pick 20 points spaced ~evenly), calculate the center and measure the distance of all points from that center.
If it is nearly a circle all points will have similar distance from the center.
If you want to do something slightly more robust, you can:
Compute center (mean) of all points.
Perform gradient descent to update the center: should be fairly easy an you won't have local minima. The error term I would probably use is max(D) - min(D) where D is the vector of distances between the blob centers and estimated circle center (but you can use robust statistics instead of max & min)
Evaluate the circle
I would use a Least Square estimation. Numerically you can use the Nelder-Mead method. You get the circle that best approximate your points and on the basis of the residual error value you decide whether to consider the circle valid or not.
Being points the array of the points, xc, yc the coordinates of the center and r the radius, this could be an example of error to minimize:
class Circle
{
private PointF[] _points;
public Circle(PointF[] points)
{
_points = points;
}
public double MinimizeFunction(double xc, double yc, double r)
{
double d, d2, dx, dy, sum;
sum = 0;
foreach(PointF p in _points)
{
dx = p.X - xc;
dy = p.Y - yc;
d2 = dx * dx + dy * dy;
// sum += d2 - r * r;
d = Math.Sqrt(d2) - r;
sum += d * d;
}
return sum;
}
public double ResidualError(double xc, double yc, double r)
{
return Math.Sqrt(MinimizeFunctional(xc, yc, r)) / (_points.Length - 3);
}
}
There is a slight difference between the commented functional and the uncommented, but for practical reason this difference is meaningless. Instead, from a theoretical point of view the difference is important.
Since you need to supply a initial values set (xc, yc, r), you can calculate the circle given three points, choosing three points far from each other.
If you need more details on "circle given three points" or Nelder-Mead you can google or ask me here.
Given two rectangles with x, y, width, height in pixels and a rotation value in degrees -- how do I calculate the closest distance of their outlines toward each other?
Background: In a game written in Lua I'm randomly generating maps, but want to ensure certain rectangles aren't too close to each other -- this is needed because maps become unsolvable if the rectangles get into certain close-distance position, as a ball needs to pass between them. Speed isn't a huge issue as I don't have many rectangles and the map is just generated once per level. Previous links I found on StackOverflow are this and this
Many thanks in advance!
Not in Lua, a Python code based on M Katz's suggestion:
def rect_distance((x1, y1, x1b, y1b), (x2, y2, x2b, y2b)):
left = x2b < x1
right = x1b < x2
bottom = y2b < y1
top = y1b < y2
if top and left:
return dist((x1, y1b), (x2b, y2))
elif left and bottom:
return dist((x1, y1), (x2b, y2b))
elif bottom and right:
return dist((x1b, y1), (x2, y2b))
elif right and top:
return dist((x1b, y1b), (x2, y2))
elif left:
return x1 - x2b
elif right:
return x2 - x1b
elif bottom:
return y1 - y2b
elif top:
return y2 - y1b
else: # rectangles intersect
return 0.
where
dist is the euclidean distance between points
rect. 1 is formed by points (x1, y1) and (x1b, y1b)
rect. 2 is formed by points (x2, y2) and (x2b, y2b)
Edit: As OK points out, this solution assumes all the rectangles are upright. To make it work for rotated rectangles as the OP asks you'd also have to compute the distance from the corners of each rectangle to the closest side of the other rectangle. But you can avoid doing that computation in most cases if the point is above or below both end points of the line segment, and to the left or right of both line segments (in telephone positions 1, 3, 7, or 9 with respect to the line segment).
Agnius's answer relies on a DistanceBetweenLineSegments() function. Here is a case analysis that does not:
(1) Check if the rects intersect. If so, the distance between them is 0.
(2) If not, think of r2 as the center of a telephone key pad, #5.
(3) r1 may be fully in one of the extreme quadrants (#1, #3, #7, or #9). If so
the distance is the distance from one rect corner to another (e.g., if r1 is
in quadrant #1, the distance is the distance from the lower-right corner of
r1 to the upper-left corner of r2).
(4) Otherwise r1 is to the left, right, above, or below r2 and the distance is
the distance between the relevant sides (e.g., if r1 is above, the distance
is the distance between r1's low y and r2's high y).
Actually there is a fast mathematical solution.
Length(Max((0, 0), Abs(Center - otherCenter) - (Extent + otherExtent)))
Where Center = ((Maximum - Minimum) / 2) + Minimum and Extent = (Maximum - Minimum) / 2.
Basically the code above zero's axis which are overlapping and therefore the distance is always correct.
It's preferable to keep the rectangle in this format as it's preferable in many situations ( a.e. rotations are much easier ).
Pseudo-code:
distance_between_rectangles = some_scary_big_number;
For each edge1 in Rectangle1:
For each edge2 in Rectangle2:
distance = calculate shortest distance between edge1 and edge2
if (distance < distance_between_rectangles)
distance_between_rectangles = distance
There are many algorithms to solve this and Agnius algorithm works fine. However I prefer the below since it seems more intuitive (you can do it on a piece of paper) and they don't rely on finding the smallest distance between lines but rather the distance between a point and a line.
The hard part is implementing the mathematical functions to find the distance between a line and a point, and to find if a point is facing a line. You can solve all this with simple trigonometry though. I have below the methodologies to do this.
For polygons (triangles, rectangles, hexagons, etc.) in arbitrary angles
If polygons overlap, return 0
Draw a line between the centres of the two polygons.
Choose the intersecting edge from each polygon. (Here we reduce the problem)
Find the smallest distance from these two edges. (You could just loop through each 4 points and look for the smallest distance to the edge of the other shape).
These algorithms work as long as any two edges of the shape don't create angles more than 180 degrees. The reason is that if something is above 180 degrees then it means that the some corners are inflated inside, like in a star.
Smallest distance between an edge and a point
If point is not facing the face, then return the smallest of the two distances between the point and the edge cornerns.
Draw a triangle from the three points (edge's points plus the solo point).
We can easily get the distances between the three drawn lines with Pythagorean Theorem.
Get the area of the triangle with Heron's formula.
Calculate the height now with Area = 12⋅base⋅height with base being the edge's length.
Check to see if a point faces an edge
As before you make a triangle from an edge and a point. Now using the Cosine law you can find all the angles with just knowing the edge distances. As long as each angle from the edge to the point is below 90 degrees, the point is facing the edge.
I have an implementation in Python for all this here if you are interested.
This question depends on what kind of distance. Do you want, distance of centers, distance of edges or distance of closest corners?
I assume you mean the last one. If the X and Y values indicate the center of the rectangle then you can find each the corners by applying this trick
//Pseudo code
Vector2 BottomLeftCorner = new Vector2(width / 2, heigth / 2);
BottomLeftCorner = BottomLeftCorner * Matrix.CreateRotation(MathHelper.ToRadians(degrees));
//If LUA has no built in Vector/Matrix calculus search for "rotate Vector" on the web.
//this helps: http://www.kirupa.com/forum/archive/index.php/t-12181.html
BottomLeftCorner += new Vector2(X, Y); //add the origin so that we have to world position.
Do this for all corners of all rectangles, then just loop over all corners and calculate the distance (just abs(v1 - v2)).
I hope this helps you
I just wrote the code for that in n-dimensions. I couldn't find a general solution easily.
// considering a rectangle object that contains two points (min and max)
double distance(const rectangle& a, const rectangle& b) const {
// whatever type you are using for points
point_type closest_point;
for (size_t i = 0; i < b.dimensions(); ++i) {
closest_point[i] = b.min[i] > a.min[i] ? a.max[i] : a.min[i];
}
// use usual euclidian distance here
return distance(a, closest_point);
}
For calculating the distance between a rectangle and a point you can:
double distance(const rectangle& a, const point_type& p) const {
double dist = 0.0;
for (size_t i = 0; i < dimensions(); ++i) {
double di = std::max(std::max(a.min[i] - p[i], p[i] - a.max[i]), 0.0);
dist += di * di;
}
return sqrt(dist);
}
If you want to rotate one of the rectangles, you need to rotate the coordinate system.
If you want to rotate both rectangles, you can rotate the coordinate system for rectangle a. Then we have to change this line:
closest_point[i] = b.min[i] > a.min[i] ? a.max[i] : a.min[i];
because this considers there is only one candidate as the closest vertex in b. You have to change it to check the distance to all vertexes in b. It's always one of the vertexes.
See: https://i.stack.imgur.com/EKJmr.png
My approach to solving the problem:
Combine the two rectangles into one large rectangle
Subtract from the large rectangle the first rectangle and the second
rectangle
What is left after the subtraction is a rectangle between the two
rectangles, the diagonal of this rectangle is the distance between
the two rectangles.
Here is an example in C#
public static double GetRectDistance(this System.Drawing.Rectangle rect1, System.Drawing.Rectangle rect2)
{
if (rect1.IntersectsWith(rect2))
{
return 0;
}
var rectUnion = System.Drawing.Rectangle.Union(rect1, rect2);
rectUnion.Width -= rect1.Width + rect2.Width;
rectUnion.Width = Math.Max(0, rectUnion.Width);
rectUnion.Height -= rect1.Height + rect2.Height;
rectUnion.Height = Math.Max(0, rectUnion.Height);
return rectUnion.Diagonal();
}
public static double Diagonal(this System.Drawing.Rectangle rect)
{
return Math.Sqrt(rect.Height * rect.Height + rect.Width * rect.Width);
}
Please check this for Java, it has the constraint all rectangles are parallel, it returns 0 for all intersecting rectangles:
public static double findClosest(Rectangle rec1, Rectangle rec2) {
double x1, x2, y1, y2;
double w, h;
if (rec1.x > rec2.x) {
x1 = rec2.x; w = rec2.width; x2 = rec1.x;
} else {
x1 = rec1.x; w = rec1.width; x2 = rec2.x;
}
if (rec1.y > rec2.y) {
y1 = rec2.y; h = rec2.height; y2 = rec1.y;
} else {
y1 = rec1.y; h = rec1.height; y2 = rec2.y;
}
double a = Math.max(0, x2 - x1 - w);
double b = Math.max(0, y2 - y1 - h);
return Math.sqrt(a*a+b*b);
}
Another solution, which calculates a number of points on the rectangle and choses the pair with the smallest distance.
Pros: works for all polygons.
Cons: a little bit less accurate and slower.
import numpy as np
import math
POINTS_PER_LINE = 100
# get points on polygon outer lines
# format of polygons: ((x1, y1), (x2, y2), ...)
def get_points_on_polygon(poly, points_per_line=POINTS_PER_LINE):
all_res = []
for i in range(len(poly)):
a = poly[i]
if i == 0:
b = poly[-1]
else:
b = poly[i-1]
res = list(np.linspace(a, b, points_per_line))
all_res += res
return all_res
# compute minimum distance between two polygons
# format of polygons: ((x1, y1), (x2, y2), ...)
def min_poly_distance(poly1, poly2, points_per_line=POINTS_PER_LINE):
poly1_points = get_points_on_polygon(poly1, points_per_line=points_per_line)
poly2_points = get_points_on_polygon(poly2, points_per_line=points_per_line)
distance = min([math.sqrt((a[0] - b[0])**2 + (a[1] - b[1])**2) for a in poly1_points for b in poly2_points])
# slower
# distance = min([np.linalg.norm(a - b) for a in poly1_points for b in poly2_points])
return distance
I have an OpenGL program (written in Delphi) that lets user draw a polygon. I want to automatically revolve (lathe) it around an axis (say, Y asix) and get a 3D shape.
How can I do this?
For simplicity, you could force at least one point to lie on the axis of rotation. You can do this easily by adding/subtracting the same value to all the x values, and the same value to all the y values, of the points in the polygon. It will retain the original shape.
The rest isn't really that hard. Pick an angle that is fairly small, say one or two degrees, and work out the coordinates of the polygon vertices as it spins around the axis. Then just join up the points with triangle fans and triangle strips.
To rotate a point around an axis is just basic Pythagoras. At 0 degrees rotation you have the points at their 2-d coordinates with a value of 0 in the third dimension.
Lets assume the points are in X and Y and we are rotating around Y. The original 'X' coordinate represents the hypotenuse. At 1 degree of rotation, we have:
sin(1) = z/hypotenuse
cos(1) = x/hypotenuse
(assuming degree-based trig functions)
To rotate a point (x, y) by angle T around the Y axis to produce a 3d point (x', y', z'):
y' = y
x' = x * cos(T)
z' = x * sin(T)
So for each point on the edge of your polygon you produce a circle of 360 points centered on the axis of rotation.
Now make a 3d shape like so:
create a GL 'triangle fan' by using your center point and the first array of rotated points
for each successive array, create a triangle strip using the points in the array and the points in the previous array
finish by creating another triangle fan centered on the center point and using the points in the last array
One thing to note is that usually, the kinds of trig functions I've used measure angles in radians, and OpenGL uses degrees. To convert degrees to radians, the formula is:
degrees = radians / pi * 180
Essentially the strategy is to sweep the profile given by the user around the given axis and generate a series of triangle strips connecting adjacent slices.
Assume that the user has drawn the polygon in the XZ plane. Further, assume that the user intends to sweep around the Z axis (i.e. the line X = 0) to generate the solid of revolution, and that one edge of the polygon lies on that axis (you can generalize later once you have this simplified case working).
For simple enough geometry, you can treat the perimeter of the polygon as a function x = f(z), that is, assume there is a unique X value for every Z value. When we go to 3D, this function becomes r = f(z), that is, the radius is unique over the length of the object.
Now, suppose we want to approximate the solid with M "slices" each spanning 2 * Pi / M radians. We'll use N "stacks" (samples in the Z dimension) as well. For each such slice, we can build a triangle strip connecting the points on one slice (i) with the points on slice (i+1). Here's some pseudo-ish code describing the process:
double dTheta = 2.0 * pi / M;
double dZ = (zMax - zMin) / N;
// Iterate over "slices"
for (int i = 0; i < M; ++i) {
double theta = i * dTheta;
double theta_next = (i+1) * dTheta;
// Iterate over "stacks":
for (int j = 0; j <= N; ++j) {
double z = zMin + i * dZ;
// Get cross-sectional radius at this Z location from your 2D model (was the
// X coordinate in the 2D polygon):
double r = f(z); // See above definition
// Convert 2D to 3D by sweeping by angle represented by this slice:
double x = r * cos(theta);
double y = r * sin(theta);
// Get coordinates of next slice over so we can join them with a triangle strip:
double xNext = r * cos(theta_next);
double yNext = r * sin(theta_next);
// Add these two points to your triangle strip (heavy pseudocode):
strip.AddPoint(x, y, z);
strip.AddPoint(xNext, yNext, z);
}
}
That's the basic idea. As sje697 said, you'll possibly need to add end caps to keep the geometry closed (i.e. a solid object, rather than a shell). But this should give you enough to get you going. This can easily be generalized to toroidal shapes as well (though you won't have a one-to-one r = f(z) function in that case).
If you just want it to rotate, then:
glRotatef(angle,0,1,0);
will rotate it around the Y-axis. If you want a lathe, then this is far more complex.
Suppose I have a map, for example from openstreetmaps.org.
I know the WGS-84 lat/lon of the upper left and lower right corner of the map.
How can I find other positions on the map from given WGS-84 lat/lon coordinates?
If the map is roughly street/city level, uses a mercator projection (as openstreetmap.org seems to), and isn't too close to the poles, linear interpolation may be accurate enough. Assuming the following:
TL = lat/lon of top left corner
BR = lat/lon of bottom right corner
P = lat/lon of the point you want to locate on the map
(w,h) = width and height of the map you have (pixels?)
the origin of the map image, (0,0), is at its top-left corner
, we could interpolate the (x,y) position corresponding to P as:
x = w * (P.lon - TL.lon) / (BR.lon - TL.lon)
y = h * (P.lat - TL.lat) / (BR.lat - TL.lat)
Common gotcha's:
The lat/lon notation convention lists the latitude first and the longitude second, i.e. "vertical" before "horizontal". This is opposite to the common x,y notation of image coordinates.
Latitude values increase when going in a north-ward direction ("up"), whereas y coordinates in your map image may be increasing when doing down.
If the map covers a larger area, linear interpolation will not be as accurate for latitudes. For a map that spans one degree of latitude and is in the earth's habitable zones (e.g. the bay area), the center latitude will be off by 0.2% or so, which is likely to by less than a pixel (depending on size)
If that's precise enough for your needs, you can stop here!
The more precise math for getting from P's latitude to a pixel y position would start with the mercator math. We know that for a latitude P.lat, the Y position on a projection starting at the equator would be as follows (I'll use a capital Y as unlike the y value we're looking for, Y starts at the equator and increases towards the north):
Y = k * ln((1 + sin(P.lat)) / (1 - sin(P.lat)))
The constant k depends on the vertical scaling of the map, which we may not know. Luckily, it can be deduced observing that y(TL) - y(BR) = h. That gets us:
k = h / (ln((1 + sin(TL.lat)) / (1 - sin(TL.lat))) - ln((1 + sin(BR.lat)) / (1 - sin(BR.lat))))
(yikes! that's four levels of brackets!) With k known, we now have the formula to find out the Y position of any latitude. We just need to correct for: (1) our y value starts at TL.lat, not the equator, and (2) y grows towards the south, rather than to the north. This gets us:
Y(TL.lat) = k * ln((1 + sin(TL.lat)) / (1 - sin(TL.lat)))
Y(P.lat) = k * ln((1 + sin(P.lat )) / (1 - sin(P.lat )))
y(P.lat) = -(Y(P.lat) - Y(TL.lat))
So this gets you:
x = w * (P.lon - TL.lon) / (BR.lon - TL.lon) // like before
y = -(Y(P.lat) - Y(TL.lat)) // where Y(anything) depends just on h, TL.lat and BR.lat