Distribution of intersection of $n$-sphere and hyperplane projected onto $c$

Question

Suppose $x$ is uniformly sampled from an n-sphere of radius $1$ restricted to points with $\langle a, x\rangle=b$ for a given unit vector $a$ and a constant $b$

Given a unit vector $c$, what is the distribution of $\langle c, x\rangle$? Does it have a name?

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Example for $a=(1,0,0), b=1/2, c=\frac{1}{\sqrt{3}}(1,1,1)$, empirical histogram of $\langle c, x\rangle$ can be seen below, what is the underlying form?

notebook

Answer 1

Without loss of generality you can take $a$ to be a vector along the coordinate $x_{n+1}$ such that the intersection is effectively a $(n-1)$-sphere in the remaining coordinates $x_1$ to $x_{n}$.

We can see the dot product $c \cdot x$ as composed of two parts

$$c \cdot x = \sum_{i=1}^{n+1} c_i x_i = \underbrace{\sum_{i=1}^{n} c_i x_i}_{\substack{\text{distribution of the coordinate}\\\text{of the (n-1)-sphere}\\\text{along the direction of $c$}}} + \underbrace{\vphantom{\sum_{i=1}^{n}} c_{n+1}x_{n+1}}_{\text{a constant term}}$$

This is a scaled and shifted power semi-circle distribution or scaled and shifted beta distribution.

For more about that distribution see: Does the distribution $f(x) \propto (1-x^2)^{n/2}$ have a name?

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The R-code below illustrates the scaling and shifting based on the 2-sphere example in the question

set.seed(1)
a = c(1,0,0)
b = 0.5
c = c(1,1,1)/sqrt(3)

### sample a uniform sphere
### reject cases where x*a is not close to b
rx = function(a,b,err = 0.02) {
    d = 1
    while (d>err) {
        x = rnorm(3)
        x = x/sum(x^2)^0.5
        d = abs(sum(x*a)-b)      
    }
    return(x)
}

### create a sample
xs = replicate(3*10^4,rx(a,b))

### statistics/images describing the sample
hist(xs[1,])
hist(xs[2,])
hist(xs[3,])
plot(xs[2,],xs[3,])

### draw the arrow on which
### the coordinate is projected
shape::Arrows(0,0,c[2],c[3],col = 2)


y = c %*% xs
dh = 0.05
hist(y, breaks = seq(min(y)-dh,max(y)+dh,dh), freq = 0, main = "")

### the constant term is x_1*c_1
### x1 satisfies x_1*a_1 = b
### then the constant is 

constant_term = b/a[1]*c[1]

### the remaining n-1 sphere has radius 
### sqrt(1 - x[1]^2)

r_sphere = sqrt(1-(b/a[1])^2)

### the vector 'c' in the space R^2 has length 

vec_c = sqrt(c[2]^2+c[3]^2) 

### the scaled beta distribution will range
### from -vec_c*r_sphere to +vec_c*r_sphere

scaling = vec_c*r_sphere*2

## adding this together, we can model it 
## as a scaled and shifted beta distribution

### a vector for the coordinates where we compute the density
ys = constant_term + seq(-scaling/2,scaling/2,scaling/10^3)

fs = dbeta((ys-constant_term)/scaling+0.5,0.5,0.5)/scaling

### adding the density to the plot
lines(ys, fs, lwd = 2)

title("histogram of simulations \n with added curve for scaled and shifted beta distribution density")

• upper image: The distribution of the coordinates $x_2 \dots x_{n+1}$ is a $(n-1)$-sphere with radius $\sqrt{1-(a_1/b)^2}$. The range of the beta distribution relates to this radius multiplied with the length of the vector $\lbrace c_2, \dots , c_{n+1}\rbrace$.
• lower image: a comparison of the histogram with the computed density. The shift of the beta distribution has 0.5 to get from the beta distribution to the semi-circle distribution and $c_1 a_1/b$ for the constant term in the product $c \cdot x$