diff --git a/_math.lua b/_math.lua
index a9278f4..08b8d16 100644
--- a/_math.lua
+++ b/_math.lua
@@ -2,26 +2,64 @@ local ffi = require 'ffi'
 
 local sin = math.sin
 local cos = math.cos
+local sqrt = math.sqrt
+local abs = math.abs
 
 local mat4
+local vec3
+local scalar
 
-local newmat4 = function(_t)
-   -- newByteData is zero-initialized
-   local data = love.data.newByteData(16 * 4)
-   local m = ffi.cast('float*', data:getFFIPointer())
-   value = {
-      data = data,
-      m = m,
-   }
-   setmetatable(value, mat4)
-   return value
-end
+scalar = {
+   near_equal = function(s1, s2, epsilon)
+      local d = abs(s1 - s2)
+      return d <= epsilon
+   end,
+}
+
+setmetatable(scalar, scalar)
 
 mat4 = {
-   __call = newmat4,
+   __call = function(_t)
+      -- newByteData is zero-initialized
+      local data = love.data.newByteData(16 * 4)
+      local m = ffi.cast('float*', data:getFFIPointer())
+      value = {
+         data = data,
+         m = m,
+      }
+      setmetatable(value, mat4)
+      return value
+   end,
+
+   set = function(m00, m01, m02, m03,
+                  m10, m11, m12, m13,
+                  m20, m21, m22, m23,
+                  m30, m31, m32, m33)
+      local M = mat4()
+      M.m[0 * 4 + 0] = m00
+      M.m[0 * 4 + 1] = m01
+      M.m[0 * 4 + 2] = m02
+      M.m[0 * 4 + 3] = m03
+
+      M.m[1 * 4 + 0] = m10
+      M.m[1 * 4 + 1] = m11
+      M.m[1 * 4 + 2] = m12
+      M.m[1 * 4 + 3] = m13
+
+      M.m[2 * 4 + 0] = m20
+      M.m[2 * 4 + 1] = m21
+      M.m[2 * 4 + 2] = m22
+      M.m[2 * 4 + 3] = m23
+
+      M.m[3 * 4 + 0] = m30
+      M.m[3 * 4 + 1] = m31
+      M.m[3 * 4 + 2] = m32
+      M.m[3 * 4 + 3] = m33
+      return M
+   end,
 
    identity = function()
-      local M = newmat4()
+      local M = mat4()
         M.m[0 * 4 + 0] = 1.0
       --M.m[0 * 4 + 1] = 0.0
       --M.m[0 * 4 + 2] = 0.0
@@ -45,7 +83,7 @@ mat4 = {
    end,
 
    translation = function(x, y, z)
-      local M = newmat4()
+      local M = mat4()
         M.m[0 * 4 + 0] = 1.0
       --M.m[0 * 4 + 1] = 0.0
       --M.m[0 * 4 + 2] = 0.0
@@ -72,7 +110,7 @@ mat4 = {
       local sin_angle = sin(angle)
       local cos_angle = cos(angle)
 
-      local M = newmat4()
+      local M = mat4()
         M.m[0 * 4 + 0] = 1.0
       --M.m[0 * 4 + 1] = 0.0
       --M.m[0 * 4 + 2] = 0.0
@@ -99,7 +137,7 @@ mat4 = {
       local sin_angle = sin(angle)
       local cos_angle = cos(angle)
 
-      local M = newmat4()
+      local M = mat4()
         M.m[0 * 4 + 0] = cos_angle
       --M.m[0 * 4 + 1] = 0.0
         M.m[0 * 4 + 2] = -sin_angle
@@ -126,7 +164,7 @@ mat4 = {
       local sin_angle = sin(angle)
       local cos_angle = cos(angle)
 
-      local M = newmat4()
+      local M = mat4()
         M.m[0 * 4 + 0] = cos_angle
         M.m[0 * 4 + 1] = sin_angle
       --M.m[0 * 4 + 2] = 0.0
@@ -149,13 +187,313 @@ mat4 = {
       return M
    end,
 
-   dot = function(a, b)
-      print("dot" .. tostring(a) .. tostring(b))
+   transpose = function(M)
+      local MT = mat4()
+      --MT.m[0 * 4 + 0] = M.m[0 * 4 + 0]
+        MT.m[0 * 4 + 1] = M.m[1 * 4 + 0]
+        MT.m[0 * 4 + 2] = M.m[2 * 4 + 0]
+        MT.m[0 * 4 + 3] = M.m[3 * 4 + 0]
+
+        MT.m[1 * 4 + 0] = M.m[0 * 4 + 1]
+      --MT.m[1 * 4 + 1] = M.m[1 * 4 + 1]
+        MT.m[1 * 4 + 2] = M.m[2 * 4 + 1]
+        MT.m[1 * 4 + 3] = M.m[3 * 4 + 1]
+
+        MT.m[2 * 4 + 0] = M.m[0 * 4 + 2]
+        MT.m[2 * 4 + 1] = M.m[1 * 4 + 2]
+      --MT.m[2 * 4 + 2] = M.m[2 * 4 + 2]
+        MT.m[2 * 4 + 3] = M.m[3 * 4 + 2]
+
+        MT.m[3 * 4 + 0] = M.m[0 * 4 + 3]
+        MT.m[3 * 4 + 1] = M.m[1 * 4 + 3]
+        MT.m[3 * 4 + 2] = M.m[2 * 4 + 3]
+      --MT.m[3 * 4 + 3] = M.m[3 * 4 + 3]
+      return MT
+   end,
+
+   look_to_lh = function(eye_position, eye_direction, up_direction)
+      assert(not vec3.equal(eye_direction, vec3._zero))
+      assert(not vec3.isinfinite(eye_direction))
+      assert(not vec3.equal(up_direction, vec3._zero))
+      assert(not vec3.isinfinite(up_direction))
+
+      local r2 = vec3.normalize(eye_direction)
+      local r0 = vec3.cross(up_direction, r2)
+      r0 = vec3.normalize(r0)
+
+      local r1 = vec3.cross(r2, r0)
+
+      local neg_eye_position = vec3.negate(eye_position)
+
+      local d0 = vec3.dot(r0, neg_eye_position)
+      local d1 = vec3.dot(r1, neg_eye_position)
+      local d2 = vec3.dot(r2, neg_eye_position)
+
+      local M = mat4()
+        M.m[0 * 4 + 0] = r0.f[0]
+        M.m[1 * 4 + 0] = r0.f[1]
+        M.m[2 * 4 + 0] = r0.f[2]
+        M.m[3 * 4 + 0] = d0
+
+        M.m[0 * 4 + 1] = r1.f[0]
+        M.m[1 * 4 + 1] = r1.f[1]
+        M.m[2 * 4 + 1] = r1.f[2]
+        M.m[3 * 4 + 1] = d1
+
+        M.m[0 * 4 + 2] = r2.f[0]
+        M.m[1 * 4 + 2] = r2.f[1]
+        M.m[2 * 4 + 2] = r2.f[2]
+        M.m[3 * 4 + 2] = d2
+
+      --M.m[0 * 4 + 3] = 0
+      --M.m[1 * 4 + 3] = 0
+      --M.m[2 * 4 + 3] = 0
+        M.m[3 * 4 + 3] = 1
+      return M
+   end,
+
+   look_at_lh = function(eye_position, focus_position, up_direction)
+      local eye_direction = vec3.subtract(focus_position, eye_position)
+      return mat4.look_to_lh(eye_position, eye_direction, up_direction)
+   end,
+
+   look_at_rh = function(eye_position, focus_position, up_direction)
+      local neg_eye_direction = vec3.subtract(eye_position, focus_position)
+      return mat4.look_to_lh(eye_position, neg_eye_direction, up_direction)
+   end,
+
+   perspective_rh = function(view_width, view_height, near_z, far_z)
+      assert(near_z > 0.0 and far_z > 0.0)
+      assert(not scalar.near_equal(view_width, 0.0, 0.00001))
+      assert(not scalar.near_equal(view_height, 0.0, 0.00001))
+      assert(not scalar.near_equal(far_z, near_z, 0.00001))
+
+      local two_near_z = near_z + near_z;
+      local f_range = far_z / (near_z - far_z);
+
+      local M = mat4()
+        M.m[0 * 4 + 0] = two_near_z / view_width
+      --M.m[0 * 4 + 1] = 0.0
+      --M.m[0 * 4 + 2] = 0.0
+      --M.m[0 * 4 + 3] = 0.0
+
+      --M.m[1 * 4 + 0] = 0.0
+        M.m[1 * 4 + 1] = two_near_z / view_height
+      --M.m[1 * 4 + 2] = 0.0
+      --M.m[1 * 4 + 3] = 0.0
+
+      --M.m[2 * 4 + 0] = 0.0
+      --M.m[2 * 4 + 1] = 0.0
+        M.m[2 * 4 + 2] = f_range
+        M.m[2 * 4 + 3] = -1.0
+
+      --M.m[3 * 4 + 0] = 0.0
+      --M.m[3 * 4 + 1] = 0.0
+        M.m[3 * 4 + 2] = f_range * near_z
+      --M.m[3 * 4 + 3] = 0.0
+      return M
+   end,
+
+   near_equal = function(M1, M2, epsilon)
+      local d00 = abs(M1.m[0 * 4 + 0] - M2.m[0 * 4 + 0])
+      local d01 = abs(M1.m[0 * 4 + 1] - M2.m[0 * 4 + 1])
+      local d02 = abs(M1.m[0 * 4 + 2] - M2.m[0 * 4 + 2])
+      local d03 = abs(M1.m[0 * 4 + 3] - M2.m[0 * 4 + 3])
+
+      local d10 = abs(M1.m[1 * 4 + 0] - M2.m[1 * 4 + 0])
+      local d11 = abs(M1.m[1 * 4 + 1] - M2.m[1 * 4 + 1])
+      local d12 = abs(M1.m[1 * 4 + 2] - M2.m[1 * 4 + 2])
+      local d13 = abs(M1.m[1 * 4 + 3] - M2.m[1 * 4 + 3])
+
+      local d20 = abs(M1.m[2 * 4 + 0] - M2.m[2 * 4 + 0])
+      local d21 = abs(M1.m[2 * 4 + 1] - M2.m[2 * 4 + 1])
+      local d22 = abs(M1.m[2 * 4 + 2] - M2.m[2 * 4 + 2])
+      local d23 = abs(M1.m[2 * 4 + 3] - M2.m[2 * 4 + 3])
+
+      local d30 = abs(M1.m[3 * 4 + 0] - M2.m[3 * 4 + 0])
+      local d31 = abs(M1.m[3 * 4 + 1] - M2.m[3 * 4 + 1])
+      local d32 = abs(M1.m[3 * 4 + 2] - M2.m[3 * 4 + 2])
+      local d33 = abs(M1.m[3 * 4 + 3] - M2.m[3 * 4 + 3])
+
+      return (
+         (d00 <= epsilon) and (d01 <= epsilon) and (d02 <= epsilon) and (d03 <= epsilon) and
+         (d10 <= epsilon) and (d11 <= epsilon) and (d12 <= epsilon) and (d13 <= epsilon) and
+         (d20 <= epsilon) and (d21 <= epsilon) and (d22 <= epsilon) and (d23 <= epsilon) and
+         (d30 <= epsilon) and (d31 <= epsilon) and (d32 <= epsilon) and (d33 <= epsilon)
+      )
+   end,
+
+   print = function(M)
+      for i = 0, 15 do
+         io.write(tostring(M.m[i]))
+         io.write(" ")
+         if i % 4 == 3 then
+            io.write("\n")
+         end
+      end
    end
 }
 
 setmetatable(mat4, mat4)
 
-return {
-   mat4 = mat4
+vec3 = {
+   __call = function(_t, x, y, z)
+      -- newByteData is zero-initialized
+      local data = love.data.newByteData(3 * 4)
+      local f = ffi.cast('float*', data:getFFIPointer())
+      value = {
+         data = data,
+         f = f,
+      }
+      f[0] = x or 0
+      f[1] = y or 0
+      f[2] = z or 0
+      setmetatable(value, vec3)
+      return value
+   end,
+
+   dot = function(v1, v2)
+      local value = (
+         v1.f[0] * v2.f[0] +
+         v1.f[1] * v2.f[1] +
+         v1.f[2] * v2.f[2]
+      )
+      return value
+   end,
+
+   length_sq = function(v)
+      return vec3.dot(v, v)
+   end,
+
+   reciprocal_length = function(v)
+      local result
+      result = vec3.length_sq(v)
+      result = 1.0 / sqrt(result)
+      return result
+   end,
+
+   length = function(v)
+      local result
+      result = vec3.length_sq(v)
+      result = sqrt(result)
+      return result
+   end,
+
+   add = function(v1, v2)
+      local result = vec3(
+         v1.f[0] + v2.f[0],
+         v1.f[1] + v2.f[1],
+         v1.f[2] + v2.f[2]
+      )
+      return result
+   end,
+
+   subtract = function(v1, v2)
+      local result = vec3(
+         v1.f[0] - v2.f[0],
+         v1.f[1] - v2.f[1],
+         v1.f[2] - v2.f[2]
+      )
+      return result
+   end,
+
+   multiply = function(v1, v2)
+      local result = vec3(
+         v1.f[0] * v2.f[0],
+         v1.f[1] * v2.f[1],
+         v1.f[2] * v2.f[2]
+      )
+      return result
+   end,
+
+   normalize = function(v)
+      local length = vec3.reciprocal_length(v)
+      local result = vec3(
+         v.f[0] * length,
+         v.f[1] * length,
+         v.f[2] * length
+      )
+      return result
+   end,
+
+   cross = function(v1, v2)
+      local result = vec3(
+         (v1.f[1] * v2.f[2]) - (v1.f[2] * v2.f[1]),
+         (v1.f[2] * v2.f[0]) - (v1.f[0] * v2.f[2]),
+         (v1.f[0] * v2.f[1]) - (v1.f[1] * v2.f[0])
+      )
+      return result
+   end,
+
+   negate = function(v)
+      local result = vec3(
+         -v.f[0],
+         -v.f[1],
+         -v.f[2]
+      )
+      return result
+   end,
+
+   equal = function(v1, v2)
+      return (
+         (v1.f[0] == v2.f[0]) and
+         (v1.f[1] == v2.f[1]) and
+         (v1.f[2] == v2.f[2])
+      )
+   end,
+
+   near_equal = function(v1, v2, epsilon)
+      local dx = abs(v1.f[0] - v2.f[0])
+      local dy = abs(v1.f[1] - v2.f[1])
+      local dz = abs(v1.f[2] - v2.f[2])
+      return (dx <= epsilon) and (dy <= epsilon) and (dz <= epsilon)
+   end,
+
+   isinfinite = function(v)
+      return (
+         (v.f[0] == -math.huge or v.f[0] == math.huge) or
+         (v.f[1] == -math.huge or v.f[1] == math.huge) or
+         (v.f[2] == -math.huge or v.f[2] == math.huge)
+      )
+   end,
+
+   print = function(v)
+      print(tostring(v.f[0]) .. " " .. tostring(v.f[1]) .. " " .. tostring(v.f[2]))
+   end,
+}
+
+setmetatable(vec3, vec3)
+vec3._zero = vec3(0, 0, 0)
+
+----------------------------------------------------------------------
+-- tests
+----------------------------------------------------------------------
+
+assert(vec3.dot(vec3(1, 3, -5), vec3(4, -2, -1)) == 3)
+assert(vec3.dot(vec3(1, 3, -5), vec3(1, 3, -5)) == 35)
+assert(vec3.length_sq(vec3(1, 3, -5)) == 35)
+assert(vec3.equal(vec3.multiply(vec3(1, 3, -5), vec3(2, 3, 4)), vec3(2, 9, -20)))
+assert(vec3.near_equal(vec3.normalize(vec3(1, 3, -5)), vec3(0.16903, 0.50709, -0.84515), 0.0001))
+assert(vec3.equal(vec3.cross(vec3(1, 2, 3), vec3(4, 5, 6)), vec3(-3, 6, -3)))
+
+assert(mat4.near_equal(mat4.look_to_lh(vec3(1, 2, 3),
+                                       vec3(5, 6, 7),
+                                       vec3(9, 10, 11)),
+                       mat4.set( 0.408249,  0.778499,  0.476731, 0.000000,
+                                -0.816496,  0.077850,  0.572078, 0.000000,
+                                 0.408248, -0.622799,  0.667424, 0.000000,
+                                -0.000001,  0.934199, -3.623158, 1.000000),
+                       0.00001))
+
+assert(mat4.near_equal(mat4.perspective_rh(2, 3, 4, 5),
+                       mat4.set(4.000000, 0.000000, 0.000000, 0.000000,
+                                0.000000, 2.666667, 0.000000, 0.000000,
+                                0.000000, 0.000000, -5.000000, -1.000000,
+                                0.000000, 0.000000, -20.000000, 0.000000),
+                       0.00001))
+
+return {
+   scalar = scalar,
+   mat4 = mat4,
+   vec3 = vec3,
 }
diff --git a/main.lua b/main.lua
index badddac..265e3de 100644
--- a/main.lua
+++ b/main.lua
@@ -3,6 +3,9 @@ local rotation
 local texture
 
 local ffi = require 'ffi'
+local _math = require '_math'
+local mat4 = _math.mat4
+local vec3 = _math.vec3
 
 local pixelcode = [[
     #pragma language glsl3
@@ -55,27 +58,6 @@ local vertexformat = {
    { name = 'VertexTexture', format = 'floatvec3', location = 2 },
 }
 
-function look_at()
-   local m = {
-      1.000000, 0.000000, 0.000000, 0.000000,
-      0.000000, 1.000000, 0.000000, 0.000000,
-      0.000000, 0.000000, 1.000000, -2.000000,
-      0.000000, 0.000000, 0.000000, 1.000000,
-   }
-
-   return m;
-end
-
-function projection()
-   local m = {
-      1.000000, 0.000000, 0.000000, 0.000000,
-      0.000000, 1.000000, 0.000000, 0.000000,
-      0.000000, 0.000000, -1.002002, -0.200200,
-      0.000000, 0.000000, -1.000000, 0.000000,
-   }
-   return m
-end
-
 function love.load(args)
    love.window.setMode(1024, 1024, {depth=true})
 
@@ -122,59 +104,19 @@ end
 
 local rotation = 0.0
 
-function rotate_x(t)
-   local cos = math.cos
-   local sin = math.sin
-   return {
-      1, 0, 0, 0,
-      0, cos(t), -sin(t), 0,
-      0, sin(t), cos(t), 0,
-      0, 0, 0, 1
-   }
-end
-
-function rotate_y(t)
-   local cos = math.cos
-   local sin = math.sin
-   return {
-      cos(t), 0, sin(t), 0,
-      0, 1, 0, 0,
-      -sin(t), 0, cos(t), 0,
-      0, 0, 0, 1
-   }
-end
-
-function rotate_z(t)
-   local cos = math.cos
-   local sin = math.sin
-   return {
-      cos(t), -sin(t), 0, 0,
-      sin(t), cos(t), 0, 0,
-      0, 0, 1, 0,
-      0, 0, 0, 1
-   }
-end
-
-function print_matrix(ptr)
-   for i=0,15 do
-      io.write(tostring(ptr[i]))
-      io.write(" ")
-      if i % 4 == 3 then
-         io.write("\n")
-      end
-   end
-end
-
-
-local math = require '_math'
-local mat4 = math.mat4
-
 function love.draw()
    local radius = 100
    local mx, my = love.mouse.getPosition()
 
-   shader:send("projection", projection(1, 1))
-   shader:send("view", "column", mat4.translation(0, 0, -2).data)
+   width, height = love.graphics.getDimensions()
+   shader:send("projection", "column", mat4.perspective_rh(width / width * 0.25,
+                                                           height / width * 0.25,
+                                                           0.1,
+                                                           1000.0).data)
+   shader:send("view", "column", mat4.look_at_rh(vec3(0, -2, 0),
+                                                 vec3(0, 0, 0),
+                                                 vec3(0, 0, 1)).data)
+
    shader:send("model", "column", mat4.rotation_x(rotation).data)
    shader:send("model2", "column", mat4.rotation_z(rotation * 0.5).data)
    shader:send("texture_sampler", texture)