Mathwords logoReference LibraryMathwords

Antipodal Points

Antipodal Points

Two points directly opposite each other on a sphere. That is, two points on opposite ends of a sphere's diameter. Note: For a sphere, antipodal means the same thing as diametrically opposed.

 

Sphere with point A at top-right and point B at bottom-left, connected by a diameter line, showing antipodal points.

Key Formula

If P=(x, y, z), then its antipodal point is P′=(−x, −y, −z)\text{If } P = (x,\, y,\, z), \text{ then its antipodal point is } P' = (-x,\, -y,\, -z)
Where:
  • PP = A point on a sphere centered at the origin
  • P′P' = The antipodal point of P, located at the diametrically opposite position
  • (x,y,z)(x, y, z) = Coordinates of P on the sphere

Worked Example

Problem: A sphere of radius 5 is centered at the origin. The point P = (3, 4, 0) lies on the sphere. Find its antipodal point and verify that the distance between the two points equals the diameter.
Step 1: Verify that P lies on the sphere by checking that its distance from the origin equals the radius.
32+42+02=9+16+0=25=5✓\sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16 + 0} = \sqrt{25} = 5 \checkmark
Step 2: Find the antipodal point by negating each coordinate of P.
P′=(−3, −4, 0)P' = (-3,\, -4,\, 0)
Step 3: Compute the distance between P and P'. Since they are on opposite ends of a diameter, this distance should equal 2r = 10.
d=(3−(−3))2+(4−(−4))2+(0−0)2=62+82=36+64=100=10d = \sqrt{(3 - (-3))^2 + (4 - (-4))^2 + (0 - 0)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10
Answer: The antipodal point is P' = (−3, −4, 0), and the distance between P and P' is 10, which equals the diameter of the sphere.

Another Example

Problem: A sphere of radius 13 is centered at the point C = (1, 2, 3). The point A = (6, 14, 3) lies on the sphere. Find the antipodal point of A.
Step 1: Verify A lies on the sphere by computing its distance from the center C = (1, 2, 3).
(6−1)2+(14−2)2+(3−3)2=25+144+0=169=13✓\sqrt{(6-1)^2 + (14-2)^2 + (3-3)^2} = \sqrt{25 + 144 + 0} = \sqrt{169} = 13 \checkmark
Step 2: When the sphere is not centered at the origin, the antipodal point A' is the reflection of A through the center C. Use the midpoint formula: since C is the midpoint of A and A', we have A' = 2C − A.
A′=2(1,2,3)−(6,14,3)=(2−6,  4−14,  6−3)=(−4, −10, 3)A' = 2(1, 2, 3) - (6, 14, 3) = (2 - 6,\; 4 - 14,\; 6 - 3) = (-4,\, -10,\, 3)
Step 3: Verify: the distance from A' to C should also equal 13.
(−4−1)2+(−10−2)2+(3−3)2=25+144=13✓\sqrt{(-4-1)^2 + (-10-2)^2 + (3-3)^2} = \sqrt{25 + 144} = 13 \checkmark
Answer: The antipodal point of A = (6, 14, 3) on this sphere is A' = (−4, −10, 3).

Frequently Asked Questions

What are antipodal points on Earth?
On Earth (modeled as a sphere), antipodal points are two locations directly opposite each other through the center of the planet. If you could drill a straight tunnel through the Earth's core, you would emerge at the antipodal point. For example, the North Pole and South Pole are antipodal. Most land locations on Earth have their antipodal point in the ocean.
Can antipodal points exist on a circle?
Yes. On a circle, two points are antipodal if they are at opposite ends of a diameter. For a unit circle, if one point is at angle θ, its antipodal point is at angle θ + 180°. The concept works the same way in two dimensions as it does on a sphere in three dimensions.

Antipodal Points vs. Diametrically Opposed Points

These terms mean exactly the same thing for spheres: two points on opposite ends of a diameter. 'Antipodal' comes from Greek (anti- meaning 'opposite' and pous meaning 'foot') and is used more often in geography and topology. 'Diametrically opposed' is used more broadly and can apply to circles as well, though both terms work in either context.

Why It Matters

Antipodal points appear throughout mathematics and science. In topology, the Borsuk–Ulam theorem states that any continuous function from a sphere to the plane must send at least one pair of antipodal points to the same value — this implies, for instance, that at any moment there exist two antipodal points on Earth with identical temperature and pressure. In navigation and geography, knowing your antipodal point helps with understanding great circle routes and global positioning.

Common Mistakes

Mistake: Negating coordinates to find the antipodal point even when the sphere is not centered at the origin.
Correction: The formula P' = −P only works when the center is at the origin. For a sphere centered at C, use the reflection formula: P' = 2C − P.
Mistake: Thinking antipodal points are just 'far apart' on a sphere rather than exactly opposite.
Correction: Antipodal points must be connected by a diameter — they are the maximum possible distance apart, and the line segment between them passes through the center of the sphere.

Related Terms