Parametric Line Point Calculator
Evaluate a point on a 3D line given in parametric form P = P₀ + t·d for any value of the parameter t.
How it works
P(t) = P₀ + t·d · x = x₀ + t·dx, y = y₀ + t·dy, z = z₀ + t·dz
A line in space is written as a base point P₀ plus the parameter t times a direction vector d. Sliding t moves you along the line: t = 0 sits at P₀, positive t moves in the direction of d, and negative t moves the opposite way. The distance travelled from P₀ is |t| times the length of d, so a unit direction vector makes t equal to arc length.
Worked example
With P₀ = (1, 2, 3), d = (2, −1, 4) and t = 2: x = 1 + 2·2 = 5, y = 2 + 2·(−1) = 0, z = 3 + 2·4 = 11, giving the point (5, 0, 11). Its distance from P₀ is 2·√(4 + 1 + 16) = 2√21 ≈ 9.165.
Sources: Line (geometry) — Wikipedia
Frequently asked questions
What does the parameter t control?
It positions you along the line. t = 0 is the base point P₀; each unit of t advances you by one direction vector d. Negative values move backward along the line.
How do I get a point a specific distance along the line?
Normalise the direction to unit length first, then set t equal to the distance you want — since distance from P₀ equals |t| times the length of d. This calculator reports that distance for whatever d and t you enter.
Can the direction vector be zero?
A genuine line needs a non-zero direction. If d is all zeros every t returns the same point P₀, so it describes a single point rather than a line.
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This calculator is for educational purposes. Double-check important results before acting on them.