= high; if ( arcLengths[ i ] === targetArcLength ) { return i / ( il - 1 ); } // we could get finer grain at lengths, or use simple interpolation between two points const lengthBefore = arcLengths[ i ]; const lengthAfter = arcLengths[ i + 1 ]; const segmentLength = lengthAfter - lengthBefore; // determine where we are between the 'before' and 'after' points const segmentFraction = ( targetArcLength - lengthBefore ) / segmentLength; // add that fractional amount to t const t = ( i + segmentFraction ) / ( il - 1 ); return t; } /** * Returns a unit vector tangent for the given interpolation factor. * If the derived curve does not implement its tangent derivation, * two points a small delta apart will be used to find its gradient * which seems to give a reasonable approximation. * * @param {number} t - The interpolation factor. * @param {(Vector2|Vector3)} [optionalTarget] - The optional target vector the result is written to. * @return {(Vector2|Vector3)} The tangent vector. */ getTangent( t, optionalTarget ) { const delta = 0.0001; let t1 = t - delta; let t2 = t + delta; // Capping in case of danger if ( t1 < 0 ) t1 = 0; if ( t2 > 1 ) t2 = 1; const pt1 = this.getPoint( t1 ); const pt2 = this.getPoint( t2 ); const tangent = optionalTarget || ( ( pt1.isVector2 ) ? new Vector2() : new Vector3() ); tangent.copy( pt2 ).sub( pt1 ).normalize(); return tangent; } /** * Same as {@link Curve#getTangent} but with equidistant samples. * * @param {number} u - The interpolation factor. * @param {(Vector2|Vector3)} [optionalTarget] - The optional target vector the result is written to. * @return {(Vector2|Vector3)} The tangent vector. * @see {@link Curve#getPointAt} */ getTangentAt( u, optionalTarget ) { const t = this.getUtoTmapping( u ); return this.getTangent( t, optionalTarget ); } /** * Generates the Frenet Frames. Requires a curve definition in 3D space. Used * in geometries like {@link TubeGeometry} or {@link ExtrudeGeometry}. * * @param {number} segments - The number of segments. * @param {boolean} [closed=false] - Whether the curve is closed or not. * @return {{tangents: Array, normals: Array, binormals: Array}} The Frenet Frames. */ computeFrenetFrames( segments, closed = false ) { // see http://www.cs.indiana.edu/pub/techreports/TR425.pdf const normal = new Vector3(); const tangents = []; const normals = []; const binormals = []; const vec = new Vector3(); const mat = new Matrix4(); // compute the tangent vectors for each segment on the curve for ( let i = 0; i <= segments; i ++ ) { const u = i / segments; tangents[ i ] = this.getTangentAt( u, new Vector3() ); } // select an initial normal vector perpendicular to the first tangent vector, // and in the direction of the minimum tangent xyz component normals[ 0 ] = new Vector3(); binormals[ 0 ] = new Vector3(); let min = Number.MAX_VALUE; const tx = Math.abs( tangents[ 0 ].x ); const ty = Math.abs( tangents[ 0 ].y ); const tz = Math.abs( tangents[ 0 ].z ); if ( tx <= min ) { min = tx; normal.set( 1, 0, 0 ); } if ( ty <= min ) { min = ty; normal.set( 0, 1, 0 ); } if ( tz <= min ) { normal.set( 0, 0, 1 ); } vec.crossVectors( tangents[ 0 ], normal ).normalize(); normals[ 0 ].crossVectors( tangents[ 0 ], vec ); binormals[ 0 ].crossVectors( tangents[ 0 ], normals[ 0 ] ); // compute the slowly-varying normal and binormal vectors for each segment on the curve for ( let i = 1; i <= segments; i ++ ) { normals[ i ] = normals[ i - 1 ].clone(); binormals[ i ] = binormals[ i - 1 ].clone(); vec.crossVectors( tangents[ i - 1 ], tangents[ i ] ); if ( vec.length() > Number.EPSILON ) { vec.normalize(); const theta = Math.acos( clamp( tangents[ i - 1 ].dot( tangents[ i ] ), -1, 1 ) ); // clamp for floating pt errors normals[ i ].applyMatrix4( mat.makeRotationAxis( vec, theta ) ); } binormals[ i ].crossVectors( tangents[ i ], normals[ i ] ); } // if the curve is closed, postprocess the vectors so the first and last normal vectors are the same if ( closed === true ) { let theta = Math.acos( clamp( normals[ 0 ].dot( normals[ segments ] ), -1, 1 ) ); theta /= segments; if ( tangents[ 0 ].dot( vec.crossVectors( normals[ 0 ], normals[ segments ] ) ) > 0 ) { theta = - theta; } for ( let i = 1; i <= segments; i ++ ) { // twist a little... normals[ i ].applyMatrix4( mat.makeRotationAxis( tangents[ i ], theta * i ) ); binormals[ i ].crossVectors( tangents[ i ], normals[ i ] ); } } return { tangents: tangents, normals: normals, binormals: binormals }; } /** * Returns a new curve with copied values from this instance. * * @return {Curve} A clone of this instance. *