The paper addresses the area of intersection problems in computer aided geometric design. An algorithm for computing all real points at which two planar algebraic curves (that is, curves defined by a polynomial implicit equation \(f(x,y)=0)\) intersect within a specified area is presented. The algorithm makes use of the expression of the curves in Bernstein form. The algorithm is suitable for curves of higher degree; by the author’s claim up to degree 25. The use of polar curves in computing double points and silhouette ones is reviewed, too, as well as how to compute all points on an algebraic curve with specified tangent direction. The intersection algorithm can also be used to compute the inflection points of an algebraic curve by the use of its Hessian.