math: add inversion code for 2×2 and 3×3 matrices, and transposition

code for all matrices.

src/lol/math/vector.h

@@ -1354,9 +1354,6 @@ template <typename T> struct Mat2
     inline Vec2<T>& operator[](size_t n) { return (&v0)[n]; }
     inline Vec2<T> const& operator[](size_t n) const { return (&v0)[n]; }
 
-    T det() const;
-    Mat2<T> invert() const;
-
     /* Helpers for transformation matrices */
     static Mat2<T> rotate(T angle);
 
@@ -1436,9 +1433,6 @@ template <typename T> struct Mat3
     inline Vec3<T>& operator[](size_t n) { return (&v0)[n]; }
     inline Vec3<T> const& operator[](size_t n) const { return (&v0)[n]; }
 
-    T det() const;
-    Mat3<T> invert() const;
-
     /* Helpers for transformation matrices */
     static Mat3<T> rotate(T angle, T x, T y, T z);
     static Mat3<T> rotate(T angle, Vec3<T> v);
@@ -1521,9 +1515,6 @@ template <typename T> struct Mat4
     inline Vec4<T>& operator[](size_t n) { return (&v0)[n]; }
     inline Vec4<T> const& operator[](size_t n) const { return (&v0)[n]; }
 
-    T det() const;
-    Mat4<T> invert() const;
-
     /* Helpers for transformation matrices */
     static Mat4<T> translate(T x, T y, T z);
     static Mat4<T> translate(Vec3<T> v);
@@ -1531,16 +1522,18 @@ template <typename T> struct Mat4
     static Mat4<T> rotate(T angle, Vec3<T> v);
     static Mat4<T> rotate(Quat<T> q);
 
-    static inline Mat4<T> translate(Mat4<T> mat, Vec3<T> v)
+    static inline Mat4<T> translate(Mat4<T> const &mat, Vec3<T> v)
     {
         return translate(v) * mat;
     }
 
-    static inline Mat4<T> rotate(Mat4<T> mat, T angle, Vec3<T> v)
+    static inline Mat4<T> rotate(Mat4<T> &mat, T angle, Vec3<T> v)
     {
         return rotate(angle, v) * mat;
     }
 
+    static Mat3<T> normal(Mat4<T> const &mat);
+
     /* Helpers for view matrices */
     static Mat4<T> lookat(Vec3<T> eye, Vec3<T> center, Vec3<T> up);
 
@@ -1556,37 +1549,37 @@ template <typename T> struct Mat4
     friend std::ostream &operator<<(std::ostream &stream, Mat4<U> const &m);
 #endif
 
-    inline Mat4<T> operator +(Mat4<T> const m) const
+    inline Mat4<T> operator +(Mat4<T> const &m) const
     {
         return Mat4<T>(v0 + m[0], v1 + m[1], v2 + m[2], v3 + m[3]);
     }
 
-    inline Mat4<T> operator +=(Mat4<T> const m)
+    inline Mat4<T> operator +=(Mat4<T> const &m)
     {
         return *this = *this + m;
     }
 
-    inline Mat4<T> operator -(Mat4<T> const m) const
+    inline Mat4<T> operator -(Mat4<T> const &m) const
     {
         return Mat4<T>(v0 - m[0], v1 - m[1], v2 - m[2], v3 - m[3]);
     }
 
-    inline Mat4<T> operator -=(Mat4<T> const m)
+    inline Mat4<T> operator -=(Mat4<T> const &m)
     {
         return *this = *this - m;
     }
 
-    inline Mat4<T> operator *(Mat4<T> const m) const
+    inline Mat4<T> operator *(Mat4<T> const &m) const
     {
         return Mat4<T>(*this * m[0], *this * m[1], *this * m[2], *this * m[3]);
     }
 
-    inline Mat4<T> operator *=(Mat4<T> const m)
+    inline Mat4<T> operator *=(Mat4<T> const &m)
     {
         return *this = *this * m;
     }
 
-    inline Vec4<T> operator *(Vec4<T> const m) const
+    inline Vec4<T> operator *(Vec4<T> const &m) const
     {
         Vec4<T> ret;
         for (int j = 0; j < 4; j++)
@@ -1602,6 +1595,18 @@ template <typename T> struct Mat4
     Vec4<T> v0, v1, v2, v3;
 };
 
+template<typename T> T determinant(Mat2<T> const &);
+template<typename T> T determinant(Mat3<T> const &);
+template<typename T> T determinant(Mat4<T> const &);
+
+template<typename T> Mat2<T> transpose(Mat2<T> const &);
+template<typename T> Mat3<T> transpose(Mat3<T> const &);
+template<typename T> Mat4<T> transpose(Mat4<T> const &);
+
+template<typename T> Mat2<T> inverse(Mat2<T> const &);
+template<typename T> Mat3<T> inverse(Mat3<T> const &);
+template<typename T> Mat4<T> inverse(Mat4<T> const &);
+
 /*
  * Arbitrarily-sized square matrices; for now this only supports
  * naive inversion and is used for the Remez inversion method.

src/math/vector.cpp

@@ -36,6 +36,12 @@ using namespace std;
 namespace lol
 {
 
+static inline float det2(float a, float b,
+                         float c, float d)
+{
+    return a * d - b * c;
+}
+
 static inline float det3(float a, float b, float c,
                          float d, float e, float f,
                          float g, float h, float i)
@@ -45,7 +51,20 @@ static inline float det3(float a, float b, float c,
          + c * (d * h - g * e);
 }
 
-static inline float cofact3(mat4 const &mat, int i, int j)
+static inline float cofact(mat2 const &mat, int i, int j)
+{
+    return mat[(i + 1) & 1][(j + 1) & 1] * (((i + j) & 1) ? -1.0f : 1.0f);
+}
+
+static inline float cofact(mat3 const &mat, int i, int j)
+{
+    return det2(mat[(i + 1) % 3][(j + 1) % 3],
+                mat[(i + 2) % 3][(j + 1) % 3],
+                mat[(i + 1) % 3][(j + 2) % 3],
+                mat[(i + 2) % 3][(j + 2) % 3]);
+}
+
+static inline float cofact(mat4 const &mat, int i, int j)
 {
     return det3(mat[(i + 1) & 3][(j + 1) & 3],
                 mat[(i + 2) & 3][(j + 1) & 3],
@@ -58,24 +77,91 @@ static inline float cofact3(mat4 const &mat, int i, int j)
                 mat[(i + 3) & 3][(j + 3) & 3]) * (((i + j) & 1) ? -1.0f : 1.0f);
 }
 
-template<> float mat4::det() const
+template<> float determinant(mat2 const &mat)
+{
+    return mat[0][0] * mat[1][1] - mat[0][1] * mat[1][0];
+}
+
+template<> mat2 transpose(mat2 const &mat)
+{
+    mat2 ret;
+    for (int j = 0; j < 2; j++)
+        for (int i = 0; i < 2; i++)
+            ret[j][i] = mat[i][j];
+    return ret;
+}
+
+template<> mat2 inverse(mat2 const &mat)
+{
+    mat2 ret;
+    float d = determinant(mat);
+    if (d)
+    {
+        d = 1.0f / d;
+        for (int j = 0; j < 2; j++)
+            for (int i = 0; i < 2; i++)
+                ret[j][i] = cofact(mat, i, j) * d;
+    }
+    return ret;
+}
+
+template<> float determinant(mat3 const &mat)
+{
+    return det3(mat[0][0], mat[0][1], mat[0][2],
+                mat[1][0], mat[1][1], mat[1][2],
+                mat[2][0], mat[2][1], mat[2][2]);
+}
+
+template<> mat3 transpose(mat3 const &mat)
+{
+    mat3 ret;
+    for (int j = 0; j < 3; j++)
+        for (int i = 0; i < 3; i++)
+            ret[j][i] = mat[i][j];
+    return ret;
+}
+
+template<> mat3 inverse(mat3 const &mat)
+{
+    mat3 ret;
+    float d = determinant(mat);
+    if (d)
+    {
+        d = 1.0f / d;
+        for (int j = 0; j < 3; j++)
+            for (int i = 0; i < 3; i++)
+                ret[j][i] = cofact(mat, i, j) * d;
+    }
+    return ret;
+}
+
+template<> float determinant(mat4 const &mat)
 {
     float ret = 0;
     for (int n = 0; n < 4; n++)
-        ret += (*this)[n][0] * cofact3(*this, n, 0);
+        ret += mat[n][0] * cofact(mat, n, 0);
     return ret;
 }
 
-template<> mat4 mat4::invert() const
+template<> mat4 transpose(mat4 const &mat)
 {
     mat4 ret;
-    float d = det();
+    for (int j = 0; j < 4; j++)
+        for (int i = 0; i < 4; i++)
+            ret[j][i] = mat[i][j];
+    return ret;
+}
+
+template<> mat4 inverse(mat4 const &mat)
+{
+    mat4 ret;
+    float d = determinant(mat);
     if (d)
     {
         d = 1.0f / d;
         for (int j = 0; j < 4; j++)
             for (int i = 0; i < 4; i++)
-                ret[j][i] = cofact3(*this, i, j) * d;
+                ret[j][i] = cofact(mat, i, j) * d;
     }
     return ret;
 }
@@ -120,6 +206,23 @@ template<> void quat::printf() const
     Log::Debug("[ %6.6f %6.6f %6.6f %6.6f ]\n", x, y, z, w);
 }
 
+template<> void mat2::printf() const
+{
+    mat2 const &p = *this;
+
+    Log::Debug("[ %6.6f %6.6f\n", p[0][0], p[1][0]);
+    Log::Debug("  %6.6f %6.6f ]\n", p[0][1], p[1][1]);
+}
+
+template<> void mat3::printf() const
+{
+    mat3 const &p = *this;
+
+    Log::Debug("[ %6.6f %6.6f %6.6f\n", p[0][0], p[1][0], p[2][0]);
+    Log::Debug("  %6.6f %6.6f %6.6f\n", p[0][1], p[1][1], p[2][1]);
+    Log::Debug("  %6.6f %6.6f %6.6f ]\n", p[0][2], p[1][2], p[2][2]);
+}
+
 template<> void mat4::printf() const
 {
     mat4 const &p = *this;

test/Makefile.am

@@ -22,8 +22,8 @@ noinst_PROGRAMS = quad benchsuite testsuite
 TESTS = testsuite
 
 testsuite_SOURCES = testsuite.cpp \
-    unit/vector.cpp unit/half.cpp unit/trig.cpp unit/build.cpp \
-    unit/real.cpp unit/image.cpp unit/quat.cpp unit/cmplx.cpp
+    unit/vector.cpp unit/matrix.cpp unit/half.cpp unit/trig.cpp \
+    unit/build.cpp unit/real.cpp unit/image.cpp unit/quat.cpp unit/cmplx.cpp
 testsuite_CPPFLAGS = @LOL_CFLAGS@ @PIPI_CFLAGS@
 testsuite_LDFLAGS = $(top_builddir)/src/liblol.a @LOL_LIBS@ @PIPI_LIBS@
 testsuite_DEPENDENCIES = $(top_builddir)/src/liblol.a

test/benchmark/vector.cpp

@@ -52,7 +52,7 @@ void bench_matrix(int mode)
         /* Determinant */
         timer.GetMs();
         for (size_t i = 0; i < MATRIX_TABLE_SIZE; i++)
-            pf[i] = pm[i].det();
+            pf[i] = determinant(pm[i]);
         result[1] += timer.GetMs();
 
         /* Multiply matrices */
@@ -70,7 +70,7 @@ void bench_matrix(int mode)
         /* Invert matrix */
         timer.GetMs();
         for (size_t i = 0; i < MATRIX_TABLE_SIZE; i++)
-            pm[i] = pm[i].invert();
+            pm[i] = inverse(pm[i]);
         result[4] += timer.GetMs();
     }
 

test/unit/matrix.cpp

@@ -0,0 +1,167 @@
+//
+// Lol Engine
+//
+// Copyright: (c) 2010-2012 Sam Hocevar <sam@hocevar.net>
+//   This program is free software; you can redistribute it and/or
+//   modify it under the terms of the Do What The Fuck You Want To
+//   Public License, Version 2, as published by Sam Hocevar. See
+//   http://sam.zoy.org/projects/COPYING.WTFPL for more details.
+//
+
+#if defined HAVE_CONFIG_H
+#   include "config.h"
+#endif
+
+#include "core.h"
+#include "lol/unit.h"
+
+namespace lol
+{
+
+LOLUNIT_FIXTURE(MatrixTest)
+{
+    void SetUp()
+    {
+        id2 = mat2(1.0f);
+        tri2 = mat2(vec2(1.0f, 0.0f),
+                    vec2(7.0f, 2.0f));
+        inv2 = mat2(vec2(4.0f, 3.0f),
+                    vec2(3.0f, 2.0f));
+
+        id3 = mat3(1.0f);
+        tri3 = mat3(vec3(1.0f, 0.0f, 0.0f),
+                    vec3(7.0f, 2.0f, 0.0f),
+                    vec3(1.0f, 5.0f, 3.0f));
+        inv3 = mat3(vec3(2.0f, 3.0f, 5.0f),
+                    vec3(3.0f, 2.0f, 3.0f),
+                    vec3(9.0f, 5.0f, 7.0f));
+
+        id4 = mat4(1.0f);
+        tri4 = mat4(vec4(1.0f, 0.0f, 0.0f, 0.0f),
+                    vec4(7.0f, 2.0f, 0.0f, 0.0f),
+                    vec4(1.0f, 5.0f, 3.0f, 0.0f),
+                    vec4(8.0f, 9.0f, 2.0f, 4.0f));
+        inv4 = mat4(vec4( 1.0f,  1.0f,  2.0f, -1.0f),
+                    vec4(-2.0f, -1.0f, -2.0f,  2.0f),
+                    vec4( 4.0f,  2.0f,  5.0f, -4.0f),
+                    vec4( 5.0f, -3.0f, -7.0f, -6.0f));
+    }
+
+    void TearDown() {}
+
+    LOLUNIT_TEST(Determinant)
+    {
+        float d1, d2;
+
+        d1 = determinant(tri2);
+        LOLUNIT_ASSERT_EQUAL(d1, 2.0f);
+        d2 = determinant(inv2);
+        LOLUNIT_ASSERT_EQUAL(d2, -1.0f);
+
+        d1 = determinant(tri3);
+        LOLUNIT_ASSERT_EQUAL(d1, 6.0f);
+        d2 = determinant(inv3);
+        LOLUNIT_ASSERT_EQUAL(d2, 1.0f);
+
+        d1 = determinant(tri4);
+        LOLUNIT_ASSERT_EQUAL(d1, 24.0f);
+        d2 = determinant(inv4);
+        LOLUNIT_ASSERT_EQUAL(d2, -1.0f);
+    }
+
+    LOLUNIT_TEST(Multiplication)
+    {
+        mat4 m0 = id4;
+        mat4 m1 = id4;
+        mat4 m2 = m0 * m1;
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][0], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][0], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][1], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][2], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][2], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][3], 1.0f);
+    }
+
+    LOLUNIT_TEST(Inverse2x2)
+    {
+        mat2 m0 = inv2;
+        mat2 m1 = inverse(m0);
+
+        mat2 m2 = m0 * m1;
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
+    }
+
+    LOLUNIT_TEST(Inverse3x3)
+    {
+        mat3 m0 = inv3;
+        mat3 m1 = inverse(m0);
+
+        mat3 m2 = m0 * m1;
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][0], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][1], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][2], 1.0f);
+    }
+
+    LOLUNIT_TEST(Inverse4x4)
+    {
+        mat4 m0 = inv4;
+        mat4 m1 = inverse(m0);
+
+        mat4 m2 = m0 * m1;
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][0], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][0], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][1], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][1], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][2], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][2], 1.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][2], 0.0f);
+
+        LOLUNIT_ASSERT_EQUAL(m2[0][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[1][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[2][3], 0.0f);
+        LOLUNIT_ASSERT_EQUAL(m2[3][3], 1.0f);
+    }
+
+    mat2 tri2, id2, inv2;
+    mat3 tri3, id3, inv3;
+    mat4 tri4, id4, inv4;
+};
+
+} /* namespace lol */
+

test/unit/vector.cpp

@@ -20,18 +20,7 @@ namespace lol
 
 LOLUNIT_FIXTURE(VectorTest)
 {
-    void SetUp()
-    {
-        identity = mat4(1.0f);
-        triangular = mat4(vec4(1.0f, 0.0f, 0.0f, 0.0f),
-                          vec4(7.0f, 2.0f, 0.0f, 0.0f),
-                          vec4(1.0f, 5.0f, 3.0f, 0.0f),
-                          vec4(8.0f, 9.0f, 2.0f, 4.0f));
-        invertible = mat4(vec4( 1.0f,  1.0f,  2.0f, -1.0f),
-                          vec4(-2.0f, -1.0f, -2.0f,  2.0f),
-                          vec4( 4.0f,  2.0f,  5.0f, -4.0f),
-                          vec4( 5.0f, -3.0f, -7.0f, -6.0f));
-    }
+    void SetUp() {}
 
     void TearDown() {}
 
@@ -135,71 +124,6 @@ LOLUNIT_FIXTURE(VectorTest)
         LOLUNIT_ASSERT_EQUAL(c.w, 0.0f);
         LOLUNIT_ASSERT_EQUAL(a3, a1);
     }
-
-    LOLUNIT_TEST(MatrixDeterminant)
-    {
-        float d1 = triangular.det();
-        LOLUNIT_ASSERT_EQUAL(d1, 24.0f);
-        float d2 = invertible.det();
-        LOLUNIT_ASSERT_EQUAL(d2, -1.0f);
-    }
-
-    LOLUNIT_TEST(MatrixMultiplication)
-    {
-        mat4 m0 = identity;
-        mat4 m1 = identity;
-        mat4 m2 = m0 * m1;
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][0], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][0], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][1], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][1], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][2], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][2], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][2], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][2], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][3], 1.0f);
-    }
-
-    LOLUNIT_TEST(MatrixInverse)
-    {
-        mat4 m0 = invertible;
-        mat4 m1 = m0.invert();
-
-        mat4 m2 = m0 * m1;
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][0], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][0], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][0], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][0], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][1], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][1], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][1], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][1], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][2], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][2], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][2], 1.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][2], 0.0f);
-
-        LOLUNIT_ASSERT_EQUAL(m2[0][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[1][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[2][3], 0.0f);
-        LOLUNIT_ASSERT_EQUAL(m2[3][3], 1.0f);
-    }
-
-    mat4 triangular, identity, invertible;
 };
 
 } /* namespace lol */

win32/testsuite.vcxproj

@@ -32,6 +32,7 @@
     <ClCompile Include="..\test\unit\cmplx.cpp" />
     <ClCompile Include="..\test\unit\half.cpp" />
     <ClCompile Include="..\test\unit\image.cpp" />
+    <ClCompile Include="..\test\unit\matrix.cpp" />
     <ClCompile Include="..\test\unit\quat.cpp" />
     <ClCompile Include="..\test\unit\real.cpp" />
     <ClCompile Include="..\test\unit\trig.cpp" />
@@ -60,4 +61,4 @@
   <Import Project="$(VCTargetsPath)\Microsoft.Cpp.targets" />
   <ImportGroup Label="ExtensionTargets">
   </ImportGroup>
-</Project>
+</Project>