I have a straight trendline on a semi-log chart (ie y axis in log scale, and x axis (ie time) in regular scale).
I think therefore the equation of that straight line is y = a * b^x
I want to move the line up or down though according to trigonometry but don't know if I can as the opposite side of the triangle is in log scale. I can do this fine on regular scale charts, but I dont know where to begin for semilog charts.
Anyone know how I do this or if there's an equation I can use to calculate how much i move the line up or down?
Related
The Xiaolin Wu algorithm draws an anti-aliased line between two points. The points can be at sub-pixel, i.e. non-integer coordinates. I'll assume the reader is familiar with the algorithm and just recall the important features. We loop across the major (longer) axis of the line, let's say it's the x-axis, basically proceeding column-by-column. In each column we color two pixels. The computation is equivalent to this: place a 1x1 square centered on the line, at the point whose x coordinate is the center of the the given column of pixels. Let's call it S. If we think of each pixel as a 1x1 square in the plane, we now calculate the area of intersection between S and each of the two pixels it straddles, and use those areas as the intensities with which to color each pixel.
That's nice and clear, but what is going on with the calculations for the endpoints? Because the endpoints can be at non-integer positions, they have to be treated as a special case. Here's the pseudocode from the linked Wikipedia article for handling the first endpoint x0, y0:
// handle first endpoint
xend := round(x0)
yend := y0 + gradient * (xend - x0)
xgap := rfpart(x0 + 0.5)
xpxl1 := xend // this will be used in the main loop
ypxl1 := ipart(yend)
plot(ypxl1, xpxl1, rfpart(yend) * xgap)
plot(ypxl1+1, xpxl1, fpart(yend) * xgap)
I edited out the if (steep) condition, so this is the code for the case when the slope of the line is less than 1. rfpart is 1-fpart, and fpart is the fractional part. ipart is the integer part.
I just have no idea what this calculation is supposed to be doing, and I can't find any explanations online. I can see that yend is the y-coordinate of the line above xend, and xend is the x coordinate of the pixel that the starting point (x0, y0) is inside of. Why are we even bothering to calculate yend? It's as if we're extending the line until the nearest integer x-coordinate.
I realize that we're coloring both the pixel that the endpoint is in, and the pixel either immediately above or below it, using certain intensities. I just don't understand the logic behind where those intensities come from.
With the Xiaolin Wu algorithm (and sub-pixel rendering techniques in general) we imagine that the screen is a continuous geometric plane, and each pixel is a 1x1 square region of that plane. We identify the centers of the pixels as being the points with integer coordinates.
First, we find the so-called "major axis" of the line, the axis along which the line is longest. Let's say that it's the x axis. We now loop across each one-pixel-wide column that the line passes through. For each column, we find the point on the line which is at the center of that column, i.e. such that the x-axis is an integer. We imagine there's a 1x1 square centered at that point. That square will completely fill the width of that column and will overlap two different pixels. We color each of those pixels according to the area of the overlap between the square and the pixel.
For the endpoints, we do things slightly differently: we still draw a square centered at the place where the line crosses the centerline of the column, but we cut that square off in the horizontal direction at the endpoint of the line. This is illustrated below.
This is a zoomed-in view of four pixels. The black crosses represent the centers of those pixels, and the red line is the line we want to draw. The red circle (x0, y0) is the starting point for the line, the line should extend from that point off to the right.
You can see the grey squares centered on the red crosses. Each pixel is going to be colored according to the area of overlap with those squares. However, in the left-hand column, we cut-off the square at x-coordinate x0. In light grey you can see the entire square, but only the part in dark grey is used for the area calculation. There are probably other ways we could have handled the endpoints, for instance we could have shifted the dark grey region up a bit so it's vertically centered at the y-coordinate y0. Presumably it doesn't make much visible difference, and this is computationally efficient.
I've annotated the drawing using the names of variables from the pseudocode on Wikipedia.
The algorithm is approximate at endpoints. This is justified because exact computation would be fairly complex (and depend on the type of endpoint), for a result barely perceivable. What matters is aliasing along the segment.
If I use euler angles, I'm aware of them having different ways to represent the same rotation but, at the same time, I need to be able to check the rotation progress in X axis. It starts, for instance, at rotation 15 degrees in X axis and I don't care about Y and Z. If I look at it after some movements of that object, maybe it shows that X axis is different although maybe X axis didn't move at all, this is because of euler angles. With quaternions this is not supposed to happen but I don't know how to check then, the progress in the rotation of one axis.
I have a chart that has several existing curves on it that I have tried to interpolate new curves in between. I have used linear interpolation in the form of y = ((x - x1)(y2 - y1) / (x2 - x1)) + y1, however the new curves look out of place.
You can see in the picture that every second line (from the bottom) is the interpolated line. While the very second line data points are exactly centered between the first and third data points in the y axis, the third line data points are not centered between the second and fourth y data points, making the graph look skew.
So I am thinking linear interpolation may not be what I am after here. Can someone recommend another method that would create curves between the existing ones, but replicates the same form?
Sudden changes in gradient are hard to interpolate. When you're at the point where you want an interpolated line to suddenly change gradient, there is no information from the points in close proximity that give information as to where the sudden change in gradient should occur.
To replicate the pattern, you actually need to copy the gradient of the line below then smoothly transition to the gradient of the line above. Visually we can see that it should occur half way between the change in gradients for the lines above and below on either side, but detecting the locations of those changes is not trivial.
The points where the sudden change in gradient are occurring are separated by a large change in the x-axis by only a small change in the y-axis. When calculating y-values for x-values in between the the changes in gradient you get the aberrations. I suggest trying to interpolate x-values based on y-values instead. For each curve, for each small arbitrary step in the y-axis, find/calculate the closest x-values from the curve on either side and take the average to plot your interpolation.
An unconventional approach may be a piece-meal style of interpolation. It may be possible to model the 3 regions of different gradients separately.
Start by identifying the 2 lines that would be drawn through the 2 sets of kinks, creating 3 regions of space. The vertical line would stop at the horizontal line near the bottom right corner of the graph.
For each region (and potentially for each value of x in each region) determine the gradient of the lines. When you're doing your interpolation of a new line, for each starting point (x1, y1), look at which region it falls in. Use the gradient of that region as a significant factor when determining the next point. Keep doing this until you reach a region boundary. When the interpolated point crosses into a different region, then use the gradient of that region as a significant factor to interpolate the next point.
It will be quite pointy if you did this strictly, so graph with some smoothing (or incorporate a smoothing factor using weighted averages of the gradients as you transition between regions, but this could be a whole lot of effort without necessarily closer results!)
I've made a chart with Excel 2010's "Combo" option for chart type so I can plot two data series for the same depth points. The image below shows the default, which is very close to what I want - except I would like to have the axes flipped so that the current X axis, which is depth, is displayed as the Y axis and both primary and secondary current Y axes plot as X axes. In other words, I'd like to rotate the chart area by 90 degrees clockwise. The "Switch Row/Column" doesn't do what I want (or expect) and I'm running out of both ideas and patience. Is there an easy fix? Or a hard fix?
Here's the plot as-is:
And here's a dummy plot of the end goal made by rotating the image around in Paint, in case the picture makes it clearer:
Finally, as was pointed out in the comments, the whole thing looks goofy and might be better plotted as a bar graph with two bars. I tried this as well and came away almost all set - but the gray bars plot from left to right and the blue bars plot from right to left. Seems like it should be as simple as changing the "Plot Series On" option to Primary Axis for both, but this destroys the graph.
I looked around and I think this link has instructions for what you're looking for: https://superuser.com/questions/188064/excel-chart-with-two-x-axes-horizontal-possible
I'm using NodeXL to plot a lot of points which are actual coordinates for cities.
The thing is that the way it's plotted now is that North America is on the Bottom Right but it should be on the top left like on a normal map.
It's like this for all of the points so pretty much I need to rotate the whole graph so that what's on the bottom right will be top left after transformation, and what's on the top right will be on the bottom left.
I have two columns with X and Y points as follows (for example):
X Y
6,238.2 9,896.0
6,141.9 9,896.0
I'm not sure the formula or Math behind this kind of rotation.
The graph is only positive so from (0,0) upward and outward to the right, there are no negative values on the x or y axis.
Could anyone help me out?
For the sake of an answer:
To quote #Tim Williams: Instead of plotting x and y plot (width-x) and (height-y)
Example: