Finding the Value of x that Divides Overlap between Two Curves Equally

In this article, we will explore how to find the value of $x$ that divides the overlapping area between two curves equally. This can be achieved by finding the point where the cumulative area of overlap is half of the total overlap area.

Introduction

When two curves overlap, they create an area that can be divided into equal parts using a single line. The goal of this article is to provide a step-by-step guide on how to find this dividing line using both direct and indirect methods.

Direct Method: Finding the Local Minimum of Cumulative Overlap Area

One approach to finding the dividing line directly from the data is to calculate the cumulative area of overlap for each value of $x$ and then find the local minimum of these cumulative areas. However, this method may not be accurate if the $x$ values are not sufficiently close together.

Example Code: Direct Method

# Calculate the overlap between y1 and y2
mydf$overlap = apply(mydf[,c("y1","y2")], 1, min)

# Calculate the area of overlap
mydf$overlap.area = cumsum(mydf$overlap * median(diff(mydf$x)))

# Find the x value where the cumulative overlap area is at its minimum
x0a = mydf$x[which.min(abs(mydf$overlap.area - 0.5*max(mydf$overlap.area)))]

# Plot the original curves and the dividing line (purple)
p0 = ggplot(mydf, aes(x = x)) + 
  geom_line(aes(y = y1), colour = 'blue') +
  geom_line(aes(y = y2), colour = 'red') +
  geom_area(aes(y = pmin(y1, y2)), fill = 'gray60') +
  geom_vline(xintercept=c(x0a), color="purple", linetype="solid") + 
  theme_classic()
p0

Indirect Method: Using Interpolation to Find the Dividing Line

A more accurate approach is to use interpolation to find the dividing line. This involves generating an interpolation function for the cumulative area of overlap and then finding the point where this function crosses zero.

Example Code: Indirect Method

# Calculate the overlap between y1 and y2
mydf$overlap = apply(mydf[,c("y1","y2")], 1, min)

# Calculate the area of overlap
mydf$overlap.area = cumsum(mydf$overlap * median(diff(mydf$x)))

# Subtract half the cumulative area to create an interpolation function for the midpoint
f = approxfun(mydf$x, mydf$overlap.area - 0.5*max(mydf$overlap.area))

# Use uniroot to find the x value where the interpolation function crosses zero
x0b = uniroot(f, range(mydf$x))$root

# Plot the original curves and the dividing lines (purple and green)
p0 = ggplot(mydf, aes(x = x)) + 
  geom_line(aes(y = y1), colour = 'blue') +
  geom_line(aes(y = y2), colour = 'red') +
  geom_area(aes(y = pmin(y1, y2)), fill = 'gray60') +
  geom_vline(xintercept=c(x0a,x0b), color=c("orange","darkgreen"), 
             linetype=c("solid", "dashed")) + 
  geom_line(aes(y=overlap.area), colour="green") + 
  theme_classic()
p0

Conclusion

In this article, we explored two methods for finding the value of $x$ that divides the overlapping area between two curves equally. The direct method involves calculating the cumulative area of overlap and then finding the local minimum of these areas. However, this method may not be accurate if the $x$ values are not sufficiently close together.

The indirect method uses interpolation to find the dividing line by generating an interpolation function for the cumulative area of overlap and then finding the point where this function crosses zero. This approach is more accurate but requires a deeper understanding of interpolation techniques.

Additional Considerations

When working with overlapping curves, it’s essential to consider the following factors:

• Smoothness: The dividing line should be smooth and continuous.
• Accuracy: The method used should provide accurate results.
• Computational Efficiency: The method used should be computationally efficient, especially for large datasets.

By understanding these considerations, you can choose the most suitable approach for your specific use case and achieve high-quality results.

Last modified on 2024-11-30