# How to find an intersection of curve and circle?

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I have a curve, derived from empirical data, and I can obtain a reasonable model of it. I need to identify a point (x, y) where the curve intersects a circle of known center and radius. The following code illustrates the question.

``x <- c(0.05, 0.20, 0.35, 0.50, 0.65, 0.80, 0.95,     1.10, 1.25, 1.40, 1.55, 1.70, 1.85, 2.00,     2.15, 2.30, 2.45, 2.60, 2.75, 2.90, 3.05)  y <- c(1.52, 1.44, 1.38, 1.31, 1.23, 1.15, 1.06,    0.96, 0.86, 0.76, 0.68, 0.61, 0.54, 0.47,     0.41, 0.36, 0.32, 0.29, 0.27, 0.26, 0.26)  fit <- loess(y ~ x, control = loess.control(surface = "direct")) newx <- data.frame(x = seq(0, 3, 0.01)) fitline <- predict(fit, newdata = newx) est <- data.frame(newx, fitline)  plot(x, y, type = "o",lwd = 2) lines(est, col = "blue", lwd = 2)  library(plotrix) draw.circle(x = 3, y = 0, radius = 2, nv = 1000, lty = 1, lwd = 1) ``

It's straightforward to find the intersection using functions from the `sf` package. Similar to @Onyambu, we first calculate the circle values:

``circ <- function(xc = 0, yc = 0,                  r = 1,                  n = 100){   v <- seq(0, 2 * pi, len = n)   cbind(x = xc + r * cos(v),         y = yc + r * sin(v)) }  m <- circ(xc = 3, yc = 0, r = 2) ``

Then convert your predicted values and the circle values to "simple features" (`LINESTRING`), and find their intersection:

``library(sf) int <- st_intersection(st_linestring(as.matrix(est)),                        st_linestring(m)) int # POINT (1.2091 0.8886608) ``

``plot(x, y, type = "o", lwd = 2) lines(est, col = "blue", lwd = 2) lines(m) points(int[1], int[2], col = "red", pch = 19) ``