Calculate the binormal from the tangent and the normal. The derivative of the curve at the tangent point gives the slope of the tangent line. For example, the derivative of X^2 is 2X so the slope of the tangent line to Y = X^2 at the point (1, 1) is 2(1) = 2. The formula for the line with slope 2 that goes through the point (1, 1) is Y - 1 = 2(X - 1) or Y = 2X - 1, which is the equation for the tangent line to Y = X^2 at the point (1, 1).
Find the normal to a curve at a point by finding the line through the tangent point that has the negative reciprocal of the tangent line. For example, the slope of the line Y = 2X - 1 is 2, so the slope of any line that is perpendicular to Y = 2X - 1 and in the same plane is -1/2. For example, if we know that the line goes through (1, 1), Y = (-1/2)X + 3/2. The normal to the tangent line to Y = X^2 at the point (1, 1) is Y = (-1/2)X + 3/2.
Compute the binormal tangent by finding a line through the tangent point that is perpendicular to both the tangent line and the normal line. The binormal tangent to Y = X^2 at the point (1,1) must be perpendicular to both Y = 2X -1 and (-1/2)X + 3/2 which lie in the same plane and are perpendicular to each other. The binormal tangent is the cross-product of the tangent and the normal -- the line through (1, 1) that comes out of the plane and is perpendicular to it. In other words, the binormal tangent is the set of points (1, 1, Z).