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This study presents a robust image watermarking method based on geometric modelling. In this method, nine samples of the approximation coefficient of the image blocks are utilised to construct a plane in the three-dimensional (3D) space. The authors change the dihedral angle formed between the created plane and the x-y plane for data embedding. To preserve the imperceptibility of the watermark, geometrical computations are used to minimise the embedding distortion. Maximum-likelihood detector is implemented to extract the watermark in the noisy channel at the receiver side. The authors experimentally determine the probability density function of the embedding dihedral angle for Gaussian samples. Owing to embedding in the dihedral angle between two planes, the proposed scheme has high robustness to gain attacks. In addition, by using the low-frequency components of the image blocks for data embedding, high robustness against noise and compression attacks has been achieved. Experimental results confirm the validity of the theoretical analysis given in this study and show the superiority of the method over similar techniques in this field. The proposed method is also robust to a wide range of attacks, namely Gaussian filtering, median filtering, JPEG compression, Gaussian noise and scaling.