+// Compile the given GLSL shader (typically a vertex or fragment shader)
+// and return the object number.
+GLuint compile_shader(const std::string &shader_src, GLenum type);
+
+// Print a 3x3 matrix to standard output. Useful for debugging.
+void print_3x3_matrix(const Eigen::Matrix3d &m);
+
+// Output a GLSL 3x3 matrix declaration.
+std::string output_glsl_mat3(const std::string &name, const Eigen::Matrix3d &m);
+
+// Calculate where to sample, and with what weight, if one wants to use
+// the GPU's bilinear hardware to sample w1 * x[0] + w2 * x[1].
+void combine_two_samples(float w1, float w2, float *offset, float *total_weight);
+