Suppose I have one set of points and one set of transformed points in 3D space. How many sample points in both 3D sets do I need to compute the unique affine transformation? My considerations are as follows: In 3D space, I can represent an affine transformation in homogeneous coordinates by a $4\times4$ matrix $T$ where 12 variables (9 for rotation and 3 for translation) are unknown. Therefore, I need 4 points in each set and solving $TP=P'$, where $P$ and $P'$ are the respective matrices containing the sampled point positions, would yield a unique solution. Is that correct?
2026-03-25 07:43:06.1774424586
Find unique 3D affine transformation between two sets of points
1.9k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GEOMETRY
- Point in, on or out of a circle
- Find all the triangles $ABC$ for which the perpendicular line to AB halves a line segment
- How to see line bundle on $\mathbb P^1$ intuitively?
- An underdetermined system derived for rotated coordinate system
- Asymptotes of hyperbola
- Finding the range of product of two distances.
- Constrain coordinates of a point into a circle
- Position of point with respect to hyperbola
- Length of Shadow from a lamp?
- Show that the asymptotes of an hyperbola are its tangents at infinity points
Related Questions in LINEAR-TRANSFORMATIONS
- Unbounded linear operator, projection from graph not open
- I don't understand this $\left(\left[T\right]^B_C\right)^{-1}=\left[T^{-1}\right]^C_B$
- A different way to define homomorphism.
- Linear algebra: what is the purpose of passive transformation matrix?
- Find matrix representation based on two vector transformations
- Is $A$ satisfying ${A^2} = - I$ similar to $\left[ {\begin{smallmatrix} 0&I \\ { - I}&0 \end{smallmatrix}} \right]$?
- Let $T:V\to W$ on finite dimensional vector spaces, is it possible to use the determinant to determine that $T$ is invertible.
- Basis-free proof of the fact that traceless linear maps are sums of commutators
- Assuming that A is the matrix of a linear operator F in S find the matrix B of F in R
- For what $k$ is $g_k\circ f_k$ invertible?
Related Questions in HOMOGENEOUS-SPACES
- How to use homogeneous coordinates and the projective plane to study the intersection of two lines
- coefficients of the sum of roots corresponding to a parabolic subgroup
- Is $O(k)\times O(n-k)$ closed in $SO(n)$?
- Homotopy groups for homogeneous spaces $SU(2n)/Sp(n)$ and $SU(2n)/SO(2n)$
- Real Hyperbolic Plane $\mathbb{RH}^2$ as Homogenous Space
- Cohomology and homology of $SU(N)/SO(N)$
- An example of a homogeneous, non-symmetric space
- Finite measure fundamental domain for a discrete group implies it's a lattice
- Different expressions of $\mathbf{CP}^n$
- Connected locally homogenous space which is not globally homogenous?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Affine transformation includes scaling (which is 3 scaling values + 3 degrees of freedom determining the directions of scaling). However, for rotation you need only 3 degrees of freedom. So you answer is correct: 4 points not lying at the same plane are enough.
You can think it that way: let's have points $p_0,\ldots p_3\to q_0,\ldots q_3$. Then if we can express any other point $p=p_0+\alpha(p_1-p_0)+\beta(p_2-p_0)+\gamma(p_3-p_0)$ by linearity of the affine transformation it would go to $q=q_0+\alpha(q_1-q_0)+\beta(q_2-q_0)+\gamma(q_3-q_0)$.