1. A handwriting input system for determining a designated position which is designated by a user on a display plane, the designated position being associated with a plurality of dot marks, the plurality of dot marks being a part of a great number of dot marks on the display plane and being formed with respect to a plurality of virtual points, a position of one of the plurality of dot marks being coded by the plurality of dot marks, and the handwriting input system specifying the position by reading the plurality of dot marks; the handwriting input system comprising: an image sensor configured to capture an image of the plurality of dot marks; dot mark position detecting circuitry configured to detect positions of the plurality of dot marks in an image which has been captured as positions in a first coordinate system of the image; decoding circuitry configured to generate relative position information based on the positions of the plurality of dot marks which is detected by the dot mark position detecting circuitry and positions of the plurality of virtual points, wherein a second coordinate system different from the first coordinate system is defined by the plurality of virtual points, the relative position information represents a positional relationship of a plurality of dot marks in the second coordinate system with respect to the plurality of virtual points, and the plurality of dot marks in the second coordinate system correspond to the plurality of dot marks in the first coordinate system; and position information calculating circuitry configured to calculate the position based on the relative position information; wherein the decoding circuitry is configured to: set initial values of relative positions of the plurality of dot marks with respect to the plurality of virtual points; then execute a calculation process of calculating a projective transformation matrix based on the positions of the plurality of dot marks in the first coordinate system and positions obtained by adding the relative positions to the positions of the virtual points; execute a transformation process of transforming the positions of the plurality of dot marks from the positions in the first coordinate system to the positions in the second coordinate system by use of the projective transformation matrix; execute a generation process of generating the relative position information based on the post-transformation positions of the plurality of dot marks and the positions of the virtual points; execute an update process of updating the relative positions in accordance with the relative position information; and sequentially execute the calculation process, the transformation process, the generation process and the update process in repetition until a prescribed condition is fulfilled.