diff --git a/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation2d.java b/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation2d.java
index 1612a5e5fa..10c7c123bd 100644
--- a/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation2d.java
+++ b/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation2d.java
@@ -109,6 +109,22 @@ public class Translation2d
     return Math.hypot(other.m_x - m_x, other.m_y - m_y);
   }
 
+  /**
+   * Calculates the square of the distance between two translations in 2D space. This is equivalent
+   * to squaring the result of {@link #getDistance(Translation2d)}, but avoids computing a square
+   * root.
+   *
+   * <p>The square of the distance between translations is defined as (x₂−x₁)²+(y₂−y₁)².
+   *
+   * @param other The translation to compute the squared distance to.
+   * @return The square of the distance between the two translations, in square meters.
+   */
+  public double getSquaredDistance(Translation2d other) {
+    double dx = other.m_x - m_x;
+    double dy = other.m_y - m_y;
+    return dx * dx + dy * dy;
+  }
+
   /**
    * Returns the X component of the translation.
    *
@@ -165,6 +181,16 @@ public class Translation2d
     return Math.hypot(m_x, m_y);
   }
 
+  /**
+   * Returns the squared norm, or squared distance from the origin to the translation. This is
+   * equivalent to squaring the result of {@link #getNorm()}, but avoids computing a square root.
+   *
+   * @return The squared norm of the translation, in square meters.
+   */
+  public double getSquaredNorm() {
+    return m_x * m_x + m_y * m_y;
+  }
+
   /**
    * Returns the angle this translation forms with the positive X axis.
    *
@@ -214,6 +240,30 @@ public class Translation2d
         (m_x - other.getX()) * rot.getSin() + (m_y - other.getY()) * rot.getCos() + other.getY());
   }
 
+  /**
+   * Computes the dot product between this translation and another translation in 2D space.
+   *
+   * <p>The dot product between two translations is defined as x₁x₂+y₁y₂.
+   *
+   * @param other The translation to compute the dot product with.
+   * @return The dot product between the two translations, in square meters.
+   */
+  public double dot(Translation2d other) {
+    return m_x * other.m_x + m_y * other.m_y;
+  }
+
+  /**
+   * Computes the cross product between this translation and another translation in 2D space.
+   *
+   * <p>The 2D cross product between two translations is defined as x₁y₂-x₂y₁.
+   *
+   * @param other The translation to compute the cross product with.
+   * @return The cross product between the two translations, in square meters.
+   */
+  public double cross(Translation2d other) {
+    return m_x * other.m_y - m_y * other.m_x;
+  }
+
   /**
    * Returns the sum of two translations in 2D space.
    *
diff --git a/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation3d.java b/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation3d.java
index a929661ee9..ff7eb01de6 100644
--- a/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation3d.java
+++ b/wpimath/src/main/java/edu/wpi/first/math/geometry/Translation3d.java
@@ -125,8 +125,26 @@ public class Translation3d
    * @return The distance between the two translations.
    */
   public double getDistance(Translation3d other) {
-    return Math.sqrt(
-        Math.pow(other.m_x - m_x, 2) + Math.pow(other.m_y - m_y, 2) + Math.pow(other.m_z - m_z, 2));
+    double dx = other.m_x - m_x;
+    double dy = other.m_y - m_y;
+    double dz = other.m_z - m_z;
+    return Math.sqrt(dx * dx + dy * dy + dz * dz);
+  }
+
+  /**
+   * Calculates the squared distance between two translations in 3D space. This is equivalent to
+   * squaring the result of {@link #getDistance(Translation3d)}, but avoids computing a square root.
+   *
+   * <p>The squared distance between translations is defined as (x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)².
+   *
+   * @param other The translation to compute the squared distance to.
+   * @return The squared distance between the two translations.
+   */
+  public double getSquaredDistance(Translation3d other) {
+    double dx = other.m_x - m_x;
+    double dy = other.m_y - m_y;
+    double dz = other.m_z - m_z;
+    return dx * dx + dy * dy + dz * dz;
   }
 
   /**
@@ -204,6 +222,16 @@ public class Translation3d
     return Math.sqrt(m_x * m_x + m_y * m_y + m_z * m_z);
   }
 
+  /**
+   * Returns the squared norm, or squared distance from the origin to the translation. This is
+   * equivalent to squaring the result of {@link #getNorm()}, but avoids computing a square root.
+   *
+   * @return The squared norm of the translation.
+   */
+  public double getSquaredNorm() {
+    return m_x * m_x + m_y * m_y + m_z * m_z;
+  }
+
   /**
    * Applies a rotation to the translation in 3D space.
    *
@@ -230,6 +258,35 @@ public class Translation3d
     return this.minus(other).rotateBy(rot).plus(other);
   }
 
+  /**
+   * Computes the dot product between this translation and another translation in 3D space.
+   *
+   * <p>The dot product between two translations is defined as x₁x₂+y₁y₂+z₁z₂.
+   *
+   * @param other The translation to compute the dot product with.
+   * @return The dot product between the two translations, in square meters.
+   */
+  public double dot(Translation3d other) {
+    return m_x * other.m_x + m_y * other.m_y + m_z * other.m_z;
+  }
+
+  /**
+   * Computes the cross product between this translation and another translation in 3D space. The
+   * resulting translation will be perpendicular to both translations.
+   *
+   * <p>The 3D cross product between two translations is defined as &lt;y₁z₂-y₂z₁, z₁x₂-z₂x₁,
+   * x₁y₂-x₂y₁&gt;.
+   *
+   * @param other The translation to compute the cross product with.
+   * @return The cross product between the two translations.
+   */
+  public Vector<N3> cross(Translation3d other) {
+    return VecBuilder.fill(
+        m_y * other.m_z - other.m_y * m_z,
+        m_z * other.m_x - other.m_z * m_x,
+        m_x * other.m_y - other.m_x * m_y);
+  }
+
   /**
    * Returns a Translation2d representing this Translation3d projected into the X-Y plane.
    *
diff --git a/wpimath/src/main/native/include/frc/geometry/Translation2d.h b/wpimath/src/main/native/include/frc/geometry/Translation2d.h
index a859743988..f6775ff2eb 100644
--- a/wpimath/src/main/native/include/frc/geometry/Translation2d.h
+++ b/wpimath/src/main/native/include/frc/geometry/Translation2d.h
@@ -13,6 +13,7 @@
 #include <wpi/json_fwd.h>
 
 #include "frc/geometry/Rotation2d.h"
+#include "units/area.h"
 #include "units/length.h"
 #include "units/math.h"
 
@@ -75,6 +76,23 @@ class WPILIB_DLLEXPORT Translation2d {
     return units::math::hypot(other.m_x - m_x, other.m_y - m_y);
   }
 
+  /**
+   * Calculates the square of the distance between two translations in 2D space.
+   * This is equivalent to squaring the result of Distance(Translation2d), but
+   * avoids computing a square root.
+   *
+   * The square of the distance between translations is defined as
+   * (x₂−x₁)²+(y₂−y₁)².
+   *
+   * @param other The translation to compute the squared distance to.
+   * @return The square of the distance between the two translations.
+   */
+  constexpr units::square_meter_t SquaredDistance(
+      const Translation2d& other) const {
+    return units::math::pow<2>(other.m_x - m_x) +
+           units::math::pow<2>(other.m_y - m_y);
+  }
+
   /**
    * Returns the X component of the translation.
    *
@@ -105,6 +123,17 @@ class WPILIB_DLLEXPORT Translation2d {
    */
   constexpr units::meter_t Norm() const { return units::math::hypot(m_x, m_y); }
 
+  /**
+   * Returns the squared norm, or squared distance from the origin to the
+   * translation. This is equivalent to squaring the result of Norm(), but
+   * avoids computing a square root.
+   *
+   * @return The squared norm of the translation.
+   */
+  constexpr units::square_meter_t SquaredNorm() const {
+    return units::math::pow<2>(m_x) + units::math::pow<2>(m_y);
+  }
+
   /**
    * Returns the angle this translation forms with the positive X axis.
    *
@@ -157,6 +186,32 @@ class WPILIB_DLLEXPORT Translation2d {
                 other.Y()};
   }
 
+  /**
+   * Computes the dot product between this translation and another translation
+   * in 2D space.
+   *
+   * The dot product between two translations is defined as x₁x₂+y₁y₂.
+   *
+   * @param other The translation to compute the dot product with.
+   * @return The dot product between the two translations.
+   */
+  constexpr units::square_meter_t Dot(const Translation2d& other) const {
+    return m_x * other.X() + m_y * other.Y();
+  }
+
+  /**
+   * Computes the cross product between this translation and another translation
+   * in 2D space.
+   *
+   * The 2D cross product between two translations is defined as x₁y₂-x₂y₁.
+   *
+   * @param other The translation to compute the cross product with.
+   * @return The cross product between the two translations.
+   */
+  constexpr units::square_meter_t Cross(const Translation2d& other) const {
+    return m_x * other.Y() - m_y * other.X();
+  }
+
   /**
    * Returns the sum of two translations in 2D space.
    *
diff --git a/wpimath/src/main/native/include/frc/geometry/Translation3d.h b/wpimath/src/main/native/include/frc/geometry/Translation3d.h
index b9876c05a3..166f871c59 100644
--- a/wpimath/src/main/native/include/frc/geometry/Translation3d.h
+++ b/wpimath/src/main/native/include/frc/geometry/Translation3d.h
@@ -14,6 +14,7 @@
 
 #include "frc/geometry/Rotation3d.h"
 #include "frc/geometry/Translation2d.h"
+#include "units/area.h"
 #include "units/length.h"
 #include "units/math.h"
 
@@ -96,6 +97,24 @@ class WPILIB_DLLEXPORT Translation3d {
                              units::math::pow<2>(other.m_z - m_z));
   }
 
+  /**
+   * Calculates the squared distance between two translations in 3D space. This
+   * is equivalent to squaring the result of Distance(Translation3d), but avoids
+   * computing a square root.
+   *
+   * The squared distance between translations is defined as
+   * (x₂−x₁)²+(y₂−y₁)²+(z₂−z₁)².
+   *
+   * @param other The translation to compute the squared distance to.
+   * @return The squared distance between the two translations.
+   */
+  constexpr units::square_meter_t SquaredDistance(
+      const Translation3d& other) const {
+    return units::math::pow<2>(other.m_x - m_x) +
+           units::math::pow<2>(other.m_y - m_y) +
+           units::math::pow<2>(other.m_z - m_z);
+  }
+
   /**
    * Returns the X component of the translation.
    *
@@ -135,6 +154,17 @@ class WPILIB_DLLEXPORT Translation3d {
     return units::math::sqrt(m_x * m_x + m_y * m_y + m_z * m_z);
   }
 
+  /**
+   * Returns the squared norm, or squared distance from the origin to the
+   * translation. This is equivalent to squaring the result of Norm(), but
+   * avoids computing a square root.
+   *
+   * @return The squared norm of the translation.
+   */
+  constexpr units::square_meter_t SquaredNorm() const {
+    return m_x * m_x + m_y * m_y + m_z * m_z;
+  }
+
   /**
    * Applies a rotation to the translation in 3D space.
    *
@@ -164,6 +194,38 @@ class WPILIB_DLLEXPORT Translation3d {
     return (*this - other).RotateBy(rot) + other;
   }
 
+  /**
+   * Computes the dot product between this translation and another translation
+   * in 3D space.
+   *
+   * The dot product between two translations is defined as x₁x₂+y₁y₂+z₁z₂.
+   *
+   * @param other The translation to compute the dot product with.
+   * @return The dot product between the two translations.
+   */
+  constexpr units::square_meter_t Dot(const Translation3d& other) const {
+    return m_x * other.X() + m_y * other.Y() + m_z * other.Z();
+  }
+
+  /**
+   * Computes the cross product between this translation and another
+   * translation in 3D space. The resulting translation will be perpendicular to
+   * both translations.
+   *
+   * The 3D cross product between two translations is defined as <y₁z₂-y₂z₁,
+   * z₁x₂-z₂x₁, x₁y₂-x₂y₁>.
+   *
+   * @param other The translation to compute the cross product with.
+   * @return The cross product between the two translations.
+   */
+  constexpr Eigen::Vector<units::square_meter_t, 3> Cross(
+      const Translation3d& other) const {
+    return Eigen::Vector<units::square_meter_t, 3>{
+        {m_y * other.Z() - other.Y() * m_z},
+        {m_z * other.X() - other.Z() * m_x},
+        {m_x * other.Y() - other.X() * m_y}};
+  }
+
   /**
    * Returns a Translation2d representing this Translation3d projected into the
    * X-Y plane.
diff --git a/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation2dTest.java b/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation2dTest.java
index 82ea7ab866..5514b90213 100644
--- a/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation2dTest.java
+++ b/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation2dTest.java
@@ -86,6 +86,12 @@ class Translation2dTest {
     assertEquals(Math.hypot(3.0, 5.0), one.getNorm(), kEpsilon);
   }
 
+  @Test
+  void testSquaredNorm() {
+    var one = new Translation2d(3.0, 5.0);
+    assertEquals(34.0, one.getSquaredNorm(), kEpsilon);
+  }
+
   @Test
   void testDistance() {
     var one = new Translation2d(1, 1);
@@ -93,6 +99,13 @@ class Translation2dTest {
     assertEquals(5.0 * Math.sqrt(2.0), one.getDistance(two), kEpsilon);
   }
 
+  @Test
+  void testSquaredDistance() {
+    var one = new Translation2d(1, 1);
+    var two = new Translation2d(6, 6);
+    assertEquals(50.0, one.getSquaredDistance(two), kEpsilon);
+  }
+
   @Test
   void testUnaryMinus() {
     var original = new Translation2d(-4.5, 7);
@@ -154,4 +167,18 @@ class Translation2dTest {
 
     assertEquals(vec, translation.toVector());
   }
+
+  @Test
+  void testDot() {
+    var one = new Translation2d(2.0, 3.0);
+    var two = new Translation2d(3.0, 4.0);
+    assertEquals(18.0, one.dot(two), kEpsilon);
+  }
+
+  @Test
+  void testCross() {
+    var one = new Translation2d(2.0, 3.0);
+    var two = new Translation2d(3.0, 4.0);
+    assertEquals(-1.0, one.cross(two), kEpsilon);
+  }
 }
diff --git a/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation3dTest.java b/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation3dTest.java
index 95a970ba6f..a8efcde086 100644
--- a/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation3dTest.java
+++ b/wpimath/src/test/java/edu/wpi/first/math/geometry/Translation3dTest.java
@@ -149,6 +149,12 @@ class Translation3dTest {
     assertEquals(Math.sqrt(83.0), one.getNorm(), kEpsilon);
   }
 
+  @Test
+  void testSquaredNorm() {
+    var one = new Translation3d(3.0, 5.0, 7.0);
+    assertEquals(83.0, one.getSquaredNorm(), kEpsilon);
+  }
+
   @Test
   void testDistance() {
     var one = new Translation3d(1.0, 1.0, 1.0);
@@ -156,6 +162,13 @@ class Translation3dTest {
     assertEquals(5.0 * Math.sqrt(3.0), one.getDistance(two), kEpsilon);
   }
 
+  @Test
+  void testSquaredDistance() {
+    var one = new Translation3d(1.0, 1.0, 1.0);
+    var two = new Translation3d(6.0, 6.0, 6.0);
+    assertEquals(75.0, one.getSquaredDistance(two), kEpsilon);
+  }
+
   @Test
   void testUnaryMinus() {
     var original = new Translation3d(-4.5, 7.0, 9.0);
@@ -225,4 +238,23 @@ class Translation3dTest {
     assertEquals(translation1, origin.nearest(List.of(translation1, translation2, translation3)));
     assertEquals(translation2, origin.nearest(List.of(translation4, translation2, translation3)));
   }
+
+  @Test
+  void testDot() {
+    var one = new Translation3d(1.0, 2.0, 3.0);
+    var two = new Translation3d(4.0, 5.0, 6.0);
+    assertEquals(32.0, one.dot(two));
+  }
+
+  @Test
+  void testCross() {
+    var one = new Translation3d(1.0, 2.0, 3.0);
+    var two = new Translation3d(4.0, 5.0, 6.0);
+
+    var cross = one.cross(two);
+    assertAll(
+        () -> assertEquals(-3.0, cross.get(0, 0), kEpsilon),
+        () -> assertEquals(6.0, cross.get(1, 0), kEpsilon),
+        () -> assertEquals(-3.0, cross.get(2, 0), kEpsilon));
+  }
 }
diff --git a/wpimath/src/test/native/cpp/geometry/Translation2dTest.cpp b/wpimath/src/test/native/cpp/geometry/Translation2dTest.cpp
index 294c213a3d..e44e334aec 100644
--- a/wpimath/src/test/native/cpp/geometry/Translation2dTest.cpp
+++ b/wpimath/src/test/native/cpp/geometry/Translation2dTest.cpp
@@ -68,12 +68,23 @@ TEST(Translation2dTest, Norm) {
   EXPECT_DOUBLE_EQ(std::hypot(3.0, 5.0), one.Norm().value());
 }
 
+TEST(Translation2dTest, SquaredNorm) {
+  const Translation2d one{3_m, 5_m};
+  EXPECT_DOUBLE_EQ(34.0, one.SquaredNorm().value());
+}
+
 TEST(Translation2dTest, Distance) {
   const Translation2d one{1_m, 1_m};
   const Translation2d two{6_m, 6_m};
   EXPECT_DOUBLE_EQ(5.0 * std::sqrt(2.0), one.Distance(two).value());
 }
 
+TEST(Translation2dTest, SquaredDistance) {
+  const Translation2d one{1_m, 1_m};
+  const Translation2d two{6_m, 6_m};
+  EXPECT_DOUBLE_EQ(50.0, one.SquaredDistance(two).value());
+}
+
 TEST(Translation2dTest, UnaryMinus) {
   const Translation2d original{-4.5_m, 7_m};
   const auto inverted = -original;
@@ -162,3 +173,15 @@ TEST(Translation2dTest, Constexpr) {
   static_assert(multiplied.X() == 2_m);
   static_assert(divided.Y() == 1_m);
 }
+
+TEST(Translation2dTest, Dot) {
+  const Translation2d one{2_m, 3_m};
+  const Translation2d two{3_m, 4_m};
+  EXPECT_DOUBLE_EQ(18.0, one.Dot(two).value());
+}
+
+TEST(Translation2dTest, Cross) {
+  const Translation2d one{2_m, 3_m};
+  const Translation2d two{3_m, 4_m};
+  EXPECT_DOUBLE_EQ(-1.0, one.Cross(two).value());
+}
diff --git a/wpimath/src/test/native/cpp/geometry/Translation3dTest.cpp b/wpimath/src/test/native/cpp/geometry/Translation3dTest.cpp
index b297f1217c..cdc7a6ebb7 100644
--- a/wpimath/src/test/native/cpp/geometry/Translation3dTest.cpp
+++ b/wpimath/src/test/native/cpp/geometry/Translation3dTest.cpp
@@ -114,12 +114,23 @@ TEST(Translation3dTest, Norm) {
   EXPECT_NEAR(one.Norm().value(), std::hypot(3, 5, 7), kEpsilon);
 }
 
+TEST(Translation3dTest, SquaredNorm) {
+  const Translation3d one{3_m, 5_m, 7_m};
+  EXPECT_NEAR(one.SquaredNorm().value(), 83.0, kEpsilon);
+}
+
 TEST(Translation3dTest, Distance) {
   const Translation3d one{1_m, 1_m, 1_m};
   const Translation3d two{6_m, 6_m, 6_m};
   EXPECT_NEAR(one.Distance(two).value(), 5 * std::sqrt(3), kEpsilon);
 }
 
+TEST(Translation3dTest, SquaredDistance) {
+  const Translation3d one{1_m, 1_m, 1_m};
+  const Translation3d two{6_m, 6_m, 6_m};
+  EXPECT_NEAR(one.SquaredDistance(two).value(), 75.0, kEpsilon);
+}
+
 TEST(Translation3dTest, UnaryMinus) {
   const Translation3d original{-4.5_m, 7_m, 9_m};
   const auto inverted = -original;
@@ -214,3 +225,19 @@ TEST(Translation3dTest, Nearest) {
   EXPECT_DOUBLE_EQ(nearest3.Y().value(), translation2.Y().value());
   EXPECT_DOUBLE_EQ(nearest3.Z().value(), translation2.Z().value());
 }
+
+TEST(Translation3dTest, Dot) {
+  const Translation3d one{1_m, 2_m, 3_m};
+  const Translation3d two{4_m, 5_m, 6_m};
+  EXPECT_NEAR(one.Dot(two).value(), 32.0, kEpsilon);
+}
+
+TEST(Translation3dTest, Cross) {
+  const Translation3d one{1_m, 2_m, 3_m};
+  const Translation3d two{4_m, 5_m, 6_m};
+
+  auto cross = one.Cross(two);
+  EXPECT_NEAR(cross[0].value(), -3.0, kEpsilon);
+  EXPECT_NEAR(cross[1].value(), 6.0, kEpsilon);
+  EXPECT_NEAR(cross[2].value(), -3.0, kEpsilon);
+}