////////////////////////////////////////////////////////////////////// // LibFile: trigonometry.scad // Trigonometry shortcuts for people who can't be bothered to remember // all the function relations, or silly acronyms like SOHCAHTOA. // Includes: // include // FileGroup: Math // FileSummary: Trigonometry shortcuts for when you can't recall the mnemonic SOHCAHTOA. // FileFootnotes: STD=Included in std.scad ////////////////////////////////////////////////////////////////////// // Section: 2D General Triangle Functions // Function: law_of_cosines() // Synopsis: Applies the Law of Cosines for an arbitrary triangle. // Topics: Geometry, Trigonometry, Triangles // See Also: law_of_sines() // Usage: // C = law_of_cosines(a, b, c); // c = law_of_cosines(a, b, C=); // Description: // Applies the Law of Cosines for an arbitrary triangle. Given three side lengths, returns the // angle in degrees for the corner opposite of the third side. Given two side lengths, and the // angle between them, returns the length of the third side. // Figure(2D): // stroke([[-50,0], [10,60], [50,0]], closed=true); // color("black") { // translate([ 33,35]) text(text="a", size=8, halign="center", valign="center"); // translate([ 0,-6]) text(text="b", size=8, halign="center", valign="center"); // translate([-22,35]) text(text="c", size=8, halign="center", valign="center"); // } // color("blue") { // translate([-37, 6]) text(text="A", size=8, halign="center", valign="center"); // translate([ 9,51]) text(text="B", size=8, halign="center", valign="center"); // translate([ 38, 6]) text(text="C", size=8, halign="center", valign="center"); // } // Arguments: // a = The length of the first side. // b = The length of the second side. // c = The length of the third side. // --- // C = The angle in degrees of the corner opposite of the third side. function law_of_cosines(a, b, c, C) = // Triangle Law of Cosines: // c^2 = a^2 + b^2 - 2*a*b*cos(C) assert(num_defined([c,C]) == 1, "Must give exactly one of c= or C=.") is_undef(c) ? sqrt(a*a + b*b - 2*a*b*cos(C)) : acos(constrain((a*a + b*b - c*c) / (2*a*b), -1, 1)); // Function: law_of_sines() // Synopsis: Applies the Law of Sines for an arbitrary triangle. // Topics: Geometry, Trigonometry, Triangles // See Also: law_of_cosines() // Usage: // B = law_of_sines(a, A, b); // b = law_of_sines(a, A, B=); // Description: // Applies the Law of Sines for an arbitrary triangle. Given two triangle side lengths and the // angle between them, returns the angle of the corner opposite of the second side. Given a side // length, the opposing angle, and a second angle, returns the length of the side opposite of the // second angle. // Figure(2D): // stroke([[-50,0], [10,60], [50,0]], closed=true); // color("black") { // translate([ 33,35]) text(text="a", size=8, halign="center", valign="center"); // translate([ 0,-6]) text(text="b", size=8, halign="center", valign="center"); // translate([-22,35]) text(text="c", size=8, halign="center", valign="center"); // } // color("blue") { // translate([-37, 6]) text(text="A", size=8, halign="center", valign="center"); // translate([ 9,51]) text(text="B", size=8, halign="center", valign="center"); // translate([ 38, 6]) text(text="C", size=8, halign="center", valign="center"); // } // Arguments: // a = The length of the first side. // A = The angle in degrees of the corner opposite of the first side. // b = The length of the second side. // --- // B = The angle in degrees of the corner opposite of the second side. function law_of_sines(a, A, b, B) = // Triangle Law of Sines: // a/sin(A) = b/sin(B) = c/sin(C) assert(num_defined([b,B]) == 1, "Must give exactly one of b= or B=.") let( r = a/sin(A) ) is_undef(b) ? r*sin(B) : asin(constrain(b/r, -1, 1)); // Section: 2D Right Triangle Functions // This is a set of functions to make it easier to perform trig calculations on right triangles. // In general, all these functions are named using these abbreviations: // - **hyp**: The length of the Hypotenuse. // - **adj**: The length of the side adjacent to the angle. // - **opp**: The length of the side opposite to the angle. // - **ang**: The angle size in degrees. // . // If you know two of those, and want to know the value of a third, you will need to call a // function named like `AAA_BBB_to_CCC()`. For example, if you know the length of the hypotenuse, // and the length of the side adjacent to the angle, and want to learn the length of the side // opposite to the angle, you will call `opp = hyp_adj_to_opp(hyp,adj);`. // Figure(2D): // color("brown") { // stroke([[40,0], [40,10], [50,10]]); // left(50) stroke(arc(r=37,angle=30)); // } // color("lightgreen") stroke([[-50,0], [50,60], [50,0]], closed=true); // color("black") { // translate([ 62,25]) text(text="opp", size=8, halign="center", valign="center"); // translate([ 0,-6]) text(text="adj", size=8, halign="center", valign="center"); // translate([ 0,40]) text(text="hyp", size=8, halign="center", valign="center"); // translate([-25, 5]) text(text="ang", size=7, halign="center", valign="center"); // } // Function: hyp_opp_to_adj() // Alias: opp_hyp_to_adj() // Synopsis: Returns the adjacent side length from the lengths of the hypotenuse and the opposite side. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // adj = hyp_opp_to_adj(hyp,opp); // adj = opp_hyp_to_adj(opp,hyp); // Description: // Given the lengths of the hypotenuse and opposite side of a right triangle, returns the length // of the adjacent side. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // opp = The length of the side of the right triangle that is opposite from the primary angle. // Example: // hyp = hyp_opp_to_adj(5,3); // Returns: 4 function hyp_opp_to_adj(hyp,opp) = assert(is_finite(hyp+opp) && hyp>=0 && opp>=0, "Triangle side lengths should be a positive numbers." ) sqrt(hyp*hyp-opp*opp); function opp_hyp_to_adj(opp,hyp) = hyp_opp_to_adj(hyp,opp); // Function: hyp_ang_to_adj() // Alias: ang_hyp_to_adj() // Synopsis: Returns the adjacent side length from the length of the hypotenuse and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // adj = hyp_ang_to_adj(hyp,ang); // adj = ang_hyp_to_adj(ang,hyp); // Description: // Given the length of the hypotenuse and the angle of the primary corner of a right triangle, // returns the length of the adjacent side. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // adj = hyp_ang_to_adj(8,60); // Returns: 4 function hyp_ang_to_adj(hyp,ang) = assert(is_finite(hyp) && hyp>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) hyp*cos(ang); function ang_hyp_to_adj(ang,hyp) = hyp_ang_to_adj(hyp, ang); // Function: opp_ang_to_adj() // Alias: ang_opp_to_adj() // Synopsis: Returns the adjacent side length from the length of the opposite side and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // adj = opp_ang_to_adj(opp,ang); // adj = ang_opp_to_adj(ang,opp); // Description: // Given the angle of the primary corner of a right triangle, and the length of the side opposite of it, // returns the length of the adjacent side. // Arguments: // opp = The length of the side of the right triangle that is opposite from the primary angle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // adj = opp_ang_to_adj(8,30); // Returns: 4 function opp_ang_to_adj(opp,ang) = assert(is_finite(opp) && opp>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) opp/tan(ang); function ang_opp_to_adj(ang,opp) = opp_ang_to_adj(opp,ang); // Function: hyp_adj_to_opp() // Alias: adj_hyp_to_opp() // Synopsis: Returns the opposite side length from the lengths of the hypotenuse and the adjacent side. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // opp = hyp_adj_to_opp(hyp,adj); // opp = adj_hyp_to_opp(adj,hyp); // Description: // Given the length of the hypotenuse and the adjacent side, returns the length of the opposite side. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // adj = The length of the side of the right triangle that is adjacent to the primary angle. // Example: // opp = hyp_adj_to_opp(5,4); // Returns: 3 function hyp_adj_to_opp(hyp,adj) = assert(is_finite(hyp) && hyp>=0 && is_finite(adj) && adj>=0, "Triangle side lengths should be a positive numbers." ) sqrt(hyp*hyp-adj*adj); function adj_hyp_to_opp(adj,hyp) = hyp_adj_to_opp(hyp,adj); // Function: hyp_ang_to_opp() // Alias: ang_hyp_to_opp() // Synopsis: Returns the opposite side length from the length of the hypotenuse and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // opp = hyp_ang_to_opp(hyp,ang); // opp = ang_hyp_to_opp(ang,hyp); // Description: // Given the length of the hypotenuse of a right triangle, and the angle of the corner, returns the length of the opposite side. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // opp = hyp_ang_to_opp(8,30); // Returns: 4 function hyp_ang_to_opp(hyp,ang) = assert(is_finite(hyp)&&hyp>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) hyp*sin(ang); function ang_hyp_to_opp(ang,hyp) = hyp_ang_to_opp(hyp,ang); // Function: adj_ang_to_opp() // Alias: ang_adj_to_opp() // Synopsis: Returns the opposite side length from the length of the adjacent side and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // opp = adj_ang_to_opp(adj,ang); // opp = ang_adj_to_opp(ang,adj); // Description: // Given the length of the adjacent side of a right triangle, and the angle of the corner, returns the length of the opposite side. // Arguments: // adj = The length of the side of the right triangle that is adjacent to the primary angle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // opp = adj_ang_to_opp(8,45); // Returns: 8 function adj_ang_to_opp(adj,ang) = assert(is_finite(adj)&&adj>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) adj*tan(ang); function ang_adj_to_opp(ang,adj) = adj_ang_to_opp(adj,ang); // Function: adj_opp_to_hyp() // Alias: opp_adj_to_hyp() // Synopsis: Returns the hypotenuse length from the lengths of the adjacent and opposite sides. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // hyp = adj_opp_to_hyp(adj,opp); // hyp = opp_adj_to_hyp(opp,adj); // Description: // Given the length of the adjacent and opposite sides of a right triangle, returns the length of the hypotenuse. // Arguments: // adj = The length of the side of the right triangle that is adjacent to the primary angle. // opp = The length of the side of the right triangle that is opposite from the primary angle. // Example: // hyp = adj_opp_to_hyp(3,4); // Returns: 5 function adj_opp_to_hyp(adj,opp) = assert(is_finite(opp) && opp>=0 && is_finite(adj) && adj>=0, "Triangle side lengths should be a positive numbers." ) norm([opp,adj]); function opp_adj_to_hyp(opp,adj) = adj_opp_to_hyp(adj,opp); // Function: adj_ang_to_hyp() // Alias: ang_adj_to_hyp() // Synopsis: Returns the hypotenuse length from the length of the adjacent and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // hyp = adj_ang_to_hyp(adj,ang); // hyp = ang_adj_to_hyp(ang,adj); // Description: // For a right triangle, given the length of the adjacent side, and the corner angle, returns the length of the hypotenuse. // Arguments: // adj = The length of the side of the right triangle that is adjacent to the primary angle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // hyp = adj_ang_to_hyp(4,60); // Returns: 8 function adj_ang_to_hyp(adj,ang) = assert(is_finite(adj) && adj>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) adj/cos(ang); function ang_adj_to_hyp(ang,adj) = adj_ang_to_hyp(adj,ang); // Function: opp_ang_to_hyp() // Alias: ang_opp_to_hyp() // Synopsis: Returns the hypotenuse length from the length of the opposite side and the angle. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // hyp = opp_ang_to_hyp(opp,ang); // hyp = ang_opp_to_hyp(ang,opp); // Description: // For a right triangle, given the length of the opposite side, and the corner angle, returns the length of the hypotenuse. // Arguments: // opp = The length of the side of the right triangle that is opposite from the primary angle. // ang = The angle in degrees of the primary corner of the right triangle. // Example: // hyp = opp_ang_to_hyp(4,30); // Returns: 8 function opp_ang_to_hyp(opp,ang) = assert(is_finite(opp) && opp>=0, "Triangle side length should be a positive number." ) assert(is_finite(ang) && ang>-90 && ang<90, "The angle should be an acute angle." ) opp/sin(ang); function ang_opp_to_hyp(ang,opp) = opp_ang_to_hyp(opp,ang); // Function: hyp_adj_to_ang() // Alias: adj_hyp_to_ang() // Synopsis: Returns the angle from the lengths of the hypotenuse and the adjacent side. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // ang = hyp_adj_to_ang(hyp,adj); // ang = adj_hyp_to_ang(adj,hyp); // Description: // For a right triangle, given the lengths of the hypotenuse and the adjacent sides, returns the angle of the corner. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // adj = The length of the side of the right triangle that is adjacent to the primary angle. // Example: // ang = hyp_adj_to_ang(8,4); // Returns: 60 degrees function hyp_adj_to_ang(hyp,adj) = assert(is_finite(hyp) && hyp>0 && is_finite(adj) && adj>=0, "Triangle side lengths should be positive numbers." ) acos(adj/hyp); function adj_hyp_to_ang(adj,hyp) = hyp_adj_to_ang(hyp,adj); // Function: hyp_opp_to_ang() // Alias: opp_hyp_to_ang() // Synopsis: Returns the angle from the lengths of the hypotenuse and the opposite side. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // ang = hyp_opp_to_ang(hyp,opp); // ang = opp_hyp_to_ang(opp,hyp); // Description: // For a right triangle, given the lengths of the hypotenuse and the opposite sides, returns the angle of the corner. // Arguments: // hyp = The length of the hypotenuse of the right triangle. // opp = The length of the side of the right triangle that is opposite from the primary angle. // Example: // ang = hyp_opp_to_ang(8,4); // Returns: 30 degrees function hyp_opp_to_ang(hyp,opp) = assert(is_finite(hyp+opp) && hyp>0 && opp>=0, "Triangle side lengths should be positive numbers." ) asin(opp/hyp); function opp_hyp_to_ang(opp,hyp) = hyp_opp_to_ang(hyp,opp); // Function: adj_opp_to_ang() // Alias: opp_adj_to_ang() // Synopsis: Returns the angle from the lengths of the adjacent and opposite sides. // Topics: Geometry, Trigonometry, Triangles // See Also: adj_ang_to_hyp(), adj_ang_to_opp(), adj_opp_to_ang(), adj_opp_to_hyp(), hyp_adj_to_ang(), hyp_adj_to_opp(), hyp_ang_to_adj(), hyp_ang_to_opp(), hyp_opp_to_adj(), hyp_opp_to_ang(), opp_ang_to_adj(), opp_ang_to_hyp() // Usage: // ang = adj_opp_to_ang(adj,opp); // ang = opp_adj_to_ang(opp,adj); // Description: // For a right triangle, given the lengths of the adjacent and opposite sides, returns the angle of the corner. // Arguments: // adj = The length of the side of the right triangle that is adjacent to the primary angle. // opp = The length of the side of the right triangle that is opposite from the primary angle. // Example: // ang = adj_opp_to_ang(sqrt(3)/2,0.5); // Returns: 30 degrees function adj_opp_to_ang(adj,opp) = assert(is_finite(adj+opp) && adj>0 && opp>=0, "Triangle side lengths should be positive numbers." ) atan2(opp,adj); function opp_adj_to_ang(opp,adj) = adj_opp_to_ang(adj,opp); // vim: expandtab tabstop=4 shiftwidth=4 softtabstop=4 nowrap